diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfymy" "b/data_all_eng_slimpj/shuffled/split2/finalzzfymy" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfymy" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nA \\emph{homomorphism} from a directed graph $G$ to a directed graph $H$ is a map from the vertices of $G$ to the vertices of $H$ which maps each edge of $G$ to an edge of $H$. Two directed graphs $G$ and $H$ are called \\emph{homomorphically equivalent} if there is a homomorphism from $G$ to $H$ and from $H$ to $G$.\nThe study of the \\emph{homomorphism order}\non the class of all finite directed graphs (or short: \\emph{digraphs}), factored by homomorphic equivalence, has a long history in graph theory. It is known to have a quite complicated structure; we refer to Ne\\v{s}et\\v{r}il and Tardif~\\cite{NesetrilTardif} and the references therein. \n\nA classical topic in graph homomorphisms is the $H$-coloring problem, which is the computational problem of deciding whether a given finite digraph $G$ maps homomorphically to $H$. The computational complexity of this problem has been classified for finite undirected graphs $H$ by Hell and Ne\\v{s}et\\v{r}il~\\cite{HellNesetril} in 1990: they are either in L or NP-complete. Feder and Vardi~\\cite{FederVardi} proved that every finite-domain CSP is polynomial-time equivalent to an $H$-coloring problem for a finite \\emph{directed} graph $H$\\footnote{This result has been sharpened in~\\cite{BulinDelicJacksonNiven}.}, and they conjectured\nthat each of these problems are either in P or NP-complete. \nThis conjecture was eventually solved in 2017 by Bulatov and, independently, by Zhuk~\\cite{BulatovFVConjecture,ZhukFVConjecture}. \nHowever, other long-standing open problems about the complexity of $H$-coloring for finite digraphs $H$ remain open, for example the characterisation of when this problem is in the complexity class L, or in NL~\\cite{LinearDatalog,EgriLaroseTessonLogspace,Kazda18}. \n\nThe border between polynomial-time tractable and NP-complete $H$-colouring problems can be described in terms of \\emph{primitive positive (pp) constructions}, which is a concept that has been introduced by Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland} in the setting of general relational structures. The idea is that if $G$ has a pp construction in $H$, then, intuitively, \\emph{`$H$ can simulate $G$'}, and the $G$-coloring problem reduces (in logarithmic space) to the $H$-coloring problem. \nIn particular, $H$-coloring is NP-hard if $K_3$ has a pp construction in $H$, where $K_3$ is the clique with three vertices, by reduction from the NP-hard three-colorability problem. It follows from the proof of the dichotomy conjecture that \n otherwise $H$-coloring is in P. Note that pp constructability can also be used to\nstudy the question of which $H$-coloring problems are in L or in NL. The surprising power of pp constructions is the motivation for studying pp constructions on finite digraphs more systematically. \n\nFor digraphs $G$ and $H$ that have at least one edge, the definition of pp constructions takes the following elegant combinatorial form: \n$G$ pp constructs $H$ if there exists a digraph $K$ and $a,b \\in V(K)^d$ for some $d \\in {\\mathbb N}$ \n such that $G$ is homomorphically equivalent to the digraph with vertices $V(H)^d$ \nand where $(u,v)$ forms an edge if there is a homomorphism from $K$ to $H$ that maps $(a,b)$ to $(u,v)$. \nWe write $H \\leq G$ if $G$ has a pp construction in $H$. It can be shown that $\\leq$ is transitive (Corollary 3.10 in~\\cite{wonderland}) and so it gives rise to a partial order $\\mathfrak P_{\\Digraphs}$ on the class of all finite digraphs (where we take the liberty to identify two digraphs $G$ and $H$ if they pp construct each other). \nSince all finite digraphs have a pp construction in $K_3$ (see, e.g. [11]), it is the smallest element of the poset $\\mathfrak P_{\\Digraphs}$.\nThe poset also has a greatest element, namely the digraph\n$P_1$ with just one vertex and no edges. \nThe digraph $P_1$ has a unique greatest lower bound in $\\mathfrak P_{\\Digraphs}$, namely the digraph $P_2$ consisting of two vertices and one directed edge; this is not hard to see and will be shown in Section~\\ref{sect:result}. \n\nIn this article, we present a complete description of the greatest lower bounds of $P_2$ in $\\mathfrak P_{\\Digraphs}$; we call these digraphs \\emph{submaximal}. We also prove that every finite digraph which does not pp constructs $P_2$ is smaller than one of the submaximal digraphs (Theorem~\\ref{thm:submaximalGraphs}; also see Figure~\\ref{fig:main}). \nThe submaximal digraphs are:\n\\begin{itemize}\n \\item The directed cycles $C_p$ for $p$ prime.\n (For $k \\in {\\mathbb N}^+$, the directed cycle $C_k$ \n is defined to be the digraph \n $({\\mathbb Z}_{k};\\{(u,v) \\mid u+1=v \\mod k\\})$.) \n \\item $T_3 \\coloneqq (\\{0,1,2\\},<)$, the transitive tournament with three vertices.\n\\end{itemize}\n\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=1.3]\n \\node (0) at (2,2) {$\\P_{1} \\equiv C_{1}$};\n \\node (1) at (2,1) {$\\P_{2}$};\n \\node (20) at (0,0) {${T}_{3}$};\n \\node (21) at (1,0) {${C}_{2}$};\n \\node (22) at (2,0) {${C}_{3}$};\n \\node (23) at (3,0) {${C}_{5}$};\n \\node (24) at (4,0) {$\\dots$};\n \n \n \\node (3) at (2,-0.7) {$\\vdots$};\n \\node[rotate = 45] (3) at (0.8,-0.7) {$\\vdots$};\n \\node[rotate = -45] (3) at (3.2,-0.7) {$\\vdots$};\n \n \n \\node (4) at (2,-1.5) {$K_3$};\n \n \\path\n (0) edge (1)\n (1) edge (20)\n (1) edge (21)\n (1) edge (22)\n (1) edge (23)\n (1) edge (24)\n ;\n \n \\end{tikzpicture}\n \\caption{The pp constructability poset on finite digraphs.}\n \\label{fig:main}\n\\end{figure}\n\n\\subsection*{Related work}\nThe pp constructability poset for smooth digraphs, i.e., digraphs where every vertex has indegree at least one and outdegree at least one (digraphs without sources and sinks), has been described in~\\cite{smooth-digraphs}. \nThe pp constructability poset on general relational structures over a two-element set has been described in~\\cite{PPPoset}.\n\n\n\\section{Minor conditions} \nPrimitive positive constructability has a universal algebraic characterisation; this characterisation plays a role in our proof, so we present it here. \nIf $H = (V,E)$ is a digraph then $H^k$ denotes the $k$-th direct power of $H$, which is the digraph with vertex set $V^k$ and edges set $$\\{((u_1,\\dots,u_k),(v_1,\\dots,v_k)) \\mid (u_1,v_1) \\in E,\\dots,(u_k,v_k) \\in E\\}.$$\nA \\emph{polymorphism} of $H$ is a homomorphism $f$ from $H^k$ to $H$, for some $k \\in {\\mathbb N}$, which is called the \\emph{arity} of $f$. We write $\\ensuremath{\\operatorname{Pol}}(H)$ for the set of all polymorphisms of $H$. This set contains the projections and is closed under composition.\\footnote{Sets of operations with these properties are called \\emph{clones} in universal algebra.} An operation $f$ is called \\emph{idempotent} if $f(x,\\dots,x) = x$ for all $x \\in V$. \n\nA central topic in universal algebra are \\emph{minor conditions}. If $f \\colon V^k \\to V$ is an operation and $\\sigma \\colon \\{1,\\dots,k\\} \\to \\{1,\\dots,n\\}$ is a function,\nthen $f_\\sigma$ denotes the operation \n$$(x_1,\\dots,x_n) \\mapsto f(x_{\\sigma(1)},\\dots,x_{\\sigma(k)}),$$\nand $f_\\sigma$ is called a \\emph{minor} of $f$. \nA \\emph{minor condition} is a set $\\Sigma$ of expressions of the form $f_{\\sigma} = g_{\\tau}$ where $f$ and $g$ are function symbols ($f$ and $g$ might be the same symbol). \n\n\\begin{example}\nAn operation $f \\colon V^n \\to V$ is called \\emph{cyclic}\nif for all $x_1,\\dots,x_n \\in V$\n$$f(x_1,x_2,\\dots,x_n) = f(x_2,\\dots,x_n,x_1).$$ This condition \ncan be expressed by the minor condition \n$$\\Sigma_n \\coloneqq \\{f_{{\\operatorname{id}}} = f_{\\tau}\\}$$ where ${\\operatorname{id}}$ denotes the identity function on $\\{1,2,\\dots,n\\}$ and $\\tau$ denotes the cyclic permutation $(1,2,\\dots,n)$ on $\\{1,\\dots,n\\}$. \n\\end{example} \n\nIf a minor condition $\\Sigma$ contains several expressions, then \ndifferent expressions in $\\Sigma$ might share the same function symbols.\n\n\\begin{example}\nAn idempotent operation $f$ is called a \\emph{Maltsev operation} \nif for all $x,y \\in V$ \n\\[f(y,y,x) = f(x,x,x) = f(x,y,y) .\\] \nThis condition can be expressed by the minor condition \n\\[\\Sigma_M \\coloneqq \\{f_{\\sigma} = f_\\tau, f_{\\tau} = f_{\\rho}\\}\\]\nwhere $\\sigma, \\tau, \\rho \\colon \\{1,2,3\\} \\to \\{1,2\\}$ are given by $\\sigma(1,2,3) = (2,2,1)$, $\\tau(1,2,3) = (1,1,1)$, and \n$\\rho(1,2,3) = (1,2,2)$. \n\\end{example}\n\nA set of operations $F$ \\emph{satisfies} a minor condition $\\Sigma$ if the function symbols in $\\Sigma$ can be replaced by operations from $F$ so that all the expressions in $\\Sigma$ hold; in this case we write $F \\models \\Sigma$. \nIf $H$ is a digraph, then $\\Sigma(H)$ denotes the class of all minor conditions that are satisfied in $\\ensuremath{\\operatorname{Pol}}(H)$. \n\n\\begin{theorem}[Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland}]\\label{thm:wonderland} \nLet $G$ and $H$ be finite digraphs. Then \n\\begin{align*}\n H \\text{ pp constructs } G &&\\text{if and only if} && \\Sigma(H)\\subseteq \\Sigma(G). \n\\end{align*}\n\\end{theorem}\n\n\\section{The pp construction poset}\\label{sect:result}\nWe have already defined pp constructability for digraphs in the introduction, but present an equivalent description here which is convenient when specifying pp constructions, and which is also closer to the presentation of Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland}. \nA \\emph{primitive positive formula} is a formula $\\phi(x_1,\\dots,x_k)$ of the form\n$$ \\exists y_1,\\dots,y_n (\\psi_1 \\wedge \\cdots \\wedge \\psi_m)$$\nwhere each of the formulas $\\psi_1,\\dots,\\psi_m$ is of the form $\\bot$ (for \\emph{false}), of the form $z_1=z_2$, or of the form $E(z_1,z_2)$ where\n$z_1,z_2$ are variables from $\\{x_1,\\dots,x_k,y_1,\\dots,y_n\\}$. \n\n\n\n\\begin{definition}\nLet $H = (V,E)$ be a digraph. A digraph $G$ with vertex set $V^d$ is called a \\emph{pp power of $H$ of dimension $d$} if there exists a primitive positive formula $\\phi(x_1,\\dots,x_d,y_1,\\dots,y_d)$\nsuch that \nthe edge set of $G$ equals \n$$\\{((u_1,\\dots,u_d),(v_1,\\dots,v_d)) \\mid \\phi(u_1,\\dots,u_d,v_1,\\dots,v_d) \\text{ holds in } H\\}.$$\n\\end{definition}\nIt follows from the definitions that $H \\leq G$ if and only if $G$ is homomorphically equivalent to a pp power of $H$. \nWe write \n\\begin{itemize}\n\\item $H \\equiv G$ if $H \\leq G$ and $G \\leq H$;\n\\item \n$H < G$ if $H \\leq G$ and not $G \\leq H$. \n\\end{itemize}\n\n\\begin{lemma}\n$\\P_1$ is the greatest element of $\\mathfrak P_{\\Digraphs}$. Moreover, $\\P_1 \\equiv {C}_1$. \n\\end{lemma}\n\\begin{proof}\nLet ${ G}$ be a finite digraph. Consider the pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq \\;\\perp$. The resulting digraph has no edges and is therefore homomorphically equivalent to $\\P_1$.\nNow consider the pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq (x=y)$. \nThe resulting digraph is homomorphically equivalent to the digraph ${C}_1$ with one vertex and a loop, which implies the statement. \n\\end{proof}\n\nIn the proof of the following lemma we need the fundamental concept of \\emph{cores} from the theory of graph homomorphisms (see, e.g.,~\\cite{HNBook}). A digraph $H = (V,E)$ is called a \\emph{core} if every \\emph{endomorphism} of $H$ (i.e., every homomorphism from $H$ to $H$) is an \\emph{embedding} (i.e., an\nisomorphism between $H$ and an induced subgraph of $H$). It is easy to see that every finite digraph $H$ is homomorphically equivalent to a core digraph, and that all core digraphs $G$ that are homomorphically equivalent to $H$ are isomorphic to each other; we therefore call $G$ \\emph{the} core of $H$. \nWhen studying $\\mathfrak P_{\\Digraphs}$ we may therefore restrict our attention to core digraphs; the big advantage of cores is the following useful lemma.\n\n\\begin{lemma}[follows from Lemma 3.9 in~\\cite{wonderland}]\\label{lem:constants}\nLet $H = (V,E)$ be a finite core digraph. Then $H \\leq G$ if and only if $G$ is homomorphically equivalent to a pp power of $H$ where the primitive positive formula might additionally contain conjuncts of the form $x = c$ where $x$ is a variable and $c \\in V$ is a constant. \n\\end{lemma}\n\n\\begin{lemma}\nWe have $\\P_2 < \\P_1$. Moreover, \n$\\P_2$ is the only coatom of $\\mathfrak P_{\\Digraphs}$, \ni.e., $\\P_2$ is the unique greatest lower bound of $\\P_1$ in $\\mathfrak P_{\\Digraphs}$. \n\\end{lemma}\n\\begin{proof}\nWe have already seen that $\\P_2 \\leq \\P_1$. \nTo prove that $\\P_2 \\not \\leq P_1$, \nfirst observe that $\\P_1$ has constant polymorphisms, while $\\P_2$ does not. Let $\\Sigma_c \\coloneqq \\{f_{\\rho} = f_{\\sigma} \\}$ \nwhere $f$ is a unary function symbol, \n$\\rho \\colon \\{1\\} \\to \\{1,2\\}$ is constant 1, and $\\sigma \\colon \\{1\\} \\to \\{1,2\\}$ is constant 2. \nThen $\\ensuremath{\\operatorname{Pol}}(\\P_1) \\models \\Sigma_c$, \nbut $\\ensuremath{\\operatorname{Pol}}(\\P_2) \\not \\models \\Sigma_c$. Then (the easy direction of) \nTheorem~\\ref{thm:wonderland} implies that $\\P_1 \\leq \\P_2$ does not hold. \n\nFor the second statement, let ${ G}$ be a finite digraph such that ${ G}<\\P_1$. \nWe have to show that ${ G} \\leq \\P_2$. \nWithout loss of generality we may assume that ${ G}$ is a core. Hence, by Lemma~\\ref{lem:constants}, we can use constants in pp constructions. Note that $G$ must have at least two different vertices $u$ and $v$. The pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq (x=u) \\wedge (y=v)$ is a digraph that has exactly one edge that is not a loop and that is therefore homomorphically equivalent to $\\P_2$.\n\\end{proof}\n\nThe following theorem is our main result and will be shown in the remainder of the article; see Figure~\\ref{fig:main}. \n\n\\begin{theorem}\\label{thm:submaximalGraphs}\nThe submaximal elements of $\\mathfrak P_{\\Digraphs}$ are precisely ${T}_3$, ${C}_2$, ${C}_3$, ${C}_5$, $\\dots$ \nIf ${ G}$ is a finite digraph that does not have a pp construction in $P_2$, then ${ G}\\leq {T}_3$ or ${ G}\\leq{C}_p$ for some prime $p$.\n\\end{theorem}\n\n\\section{Submaximal digraphs and minor conditions}\nWe first discuss which of \nthe minor conditions that we have encountered \nare satisfied by the polymorphisms of \nthe digraphs that appear in Theorem~\\ref{thm:submaximalGraphs}. \nThe following facts are well-known; we present the proof for the convenience of the reader. \n\n\\begin{lemma}\\label{lem:CpConditions1}\nLet $p$ and $q$ be primes. Then\n$\\ensuremath{\\operatorname{Pol}}(C_p) \\models \\Sigma_q$ if and only if $q \\neq p$. \n\\end{lemma}\n\\begin{proof}\n If $p\\neq q$, then there is an $n\\in \\mathbb{N}^+\\!$ such that $p\\cdot n=1\\pmod q$. The map\n\\[(x_1,\\dots,x_p) \\mapsto n\\cdot(x_1 +\\ldots + x_p) \\pmod q\\]\nis a polymorphism of ${C}_q$ satisfying $\\Sigma_p$.\n\nNow suppose that $p=q$. We assume for contradiction that $f$ is a polymorphism of ${C}_q$ satisfying $\\Sigma_p$. Then\n\\[f(0,\\dots,q-2,q-1) = a = f(1,\\dots,q-1,0)\\]\nand hence $(a,a) \\in E$, which is impossible since $C_p$ has a loop only if $p=1$. \n\\end{proof}\n\n\\begin{lemma}\\label{lem:CpConditions2}\n$\\ensuremath{\\operatorname{Pol}}(C_n) \\models \\Sigma_M$ for every $n \\in {\\mathbb N}$. \n\\end{lemma}\n\\begin{proof}\nThe ternary operation $(x_1,x_2,x_3) \\mapsto x_1 - x_2 + x_3 \\pmod n$ is a Maltsev polymorphism of $C_n$. \n\\end{proof}\n\n\nA finite digraph $H$ is called \n\\emph{$k$-rectangular} if whenever $H$ contains directed paths of length $k$ from $a$ to $b$, from $c$ to $b$, \nand from $c$ to $d$, then also from $a$ to $d$. See Figure~\\ref{fig:rect}. \nA digraph $H$ is called \\emph{totally rectangular} if it is $k$-rectangular for all $k \\geq 1$. \n\n\\begin{lemma}\\label{lem:maltIffRect}\nA finite digraph $H$ is totally rectangular if and only if\nit has a Maltsev polymorphism. \nA finite core digraph $H$ has a Maltsev polymorphism if and only if \n$\\ensuremath{\\operatorname{Pol}}(H) \\models \\Sigma_M$. \n\\end{lemma}\n\\begin{proof}\nThe first part of the statement is Corollary 4.11 in~\\cite{CarvalhoEgriJacksonNiven}. For the second statement, \nlet $H = (V,E)$ be a core digraph which has a polymorphism $f$ that satisfies $f(x,y,y)=f(x,x,x)=f(y,y,x)$ for all $x,y \\in V$; we have to find a polymorphism that is additionally idempotent. \nNote that the function $x \\mapsto f(x,x,x)$ is an endomorphism; since $H$ is a core, the endomorphism is injective. Since $H$ is finite the endomorphism must in fact be an automorphism, and has an inverse $i$ which is an endomorphism as well. Then the operation $(x_1,x_2,x_3) \\mapsto i(f(x_1,x_2,x_3))$ is idempotent and a Maltsev operation. \n\\end{proof}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=1.3]\n \\node (a) at (0,0) {$a$};\n \\node (b) at (0,1) {$b$};\n \\node (c) at (1,1) {$c$};\n \\node (d) at (1,0) {$d$};\n \\path[->]\n (a) edge (b)\n \n (c) edge (b)\n (c) edge (d)\n (a) edge[dashed] (d)\n ;\n \\end{tikzpicture}\n \\caption{Rectangularity in digraphs.}\n \\label{fig:rect}\n\\end{figure}\n\n\n\n\\begin{lemma}\\label{lem:T3Conditions}\n$\\ensuremath{\\operatorname{Pol}}(T_3) \\models \\Sigma_n$ for every $n \\in {\\mathbb N}$, but $\\ensuremath{\\operatorname{Pol}}(T_3) \\not \\models \\Sigma_M$. \n\\end{lemma}\n\\begin{proof}\nThe operation $(x_1,\\dots,x_n) \\mapsto \\max(x_1,\\dots,x_n)$ is a polymorphism of $T_3$ that satisfies $\\Sigma_n$. \nOn the other hand, $T_3 = (\\{0,1,2\\},E)$ is not 1-rectangular, witnessed by $(1,2),(0,2),(0,1) \\in E$ but $(1,1) \\notin E$; the second statement therefore follows from \nLemma~\\ref{lem:maltIffRect}. \n\\end{proof}\n\n\nThe following theorem states that \nthe digraph $\\P_2$ is the unique smallest element of \n$\\mathfrak P_{\\Digraphs}$ that satisfies $\\Sigma_M$ and $\\Sigma_p$ for all $p$ prime. \n\n\\begin{theorem}\\label{thm:maltAndCyclImplyIdempotent}\nLet ${ G}$ be a finite digraph\nthat satisfies $\\Sigma_M$\nand $\\Sigma_p$ for all primes $p$. Then \n$\\P_2 \\leq { G}$. \n\\end{theorem}\n\n\nIn the proof of Theorem~\\ref{thm:maltAndCyclImplyIdempotent} we make use the following result of Carvalho, Egri, Jackson, and Niven~\\cite{CarvalhoEgriJacksonNiven}, which guides us in our further proof steps. \n\n\\begin{theorem}[Lemma 3.10 in~\\cite{CarvalhoEgriJacksonNiven}]\\label{thm:maltImpliesPathOrDuoc}\nIf ${ G}$ is totally rectangular, then ${ G}$ is homomorphically equivalent to either a directed path or a disjoint union of directed cycles.\n\\end{theorem}\n\nWe write $\\P_k$ for the directed path with the $k$ vertices $\\{0,\\dots,k-1\\}$. \n\n\\begin{lemma}\\label{lem:IdempConstructPaths}\nThe digraph $\\P_2$ pp constructs $\\P_{k}$ for all $k\\in\\mathbb{N}^+$\\!.\n\\end{lemma}\n\\begin{proof}\nClearly, $\\P_2 \\leq \\P_1$ and $\\P_2 \\leq \\P_2$.\nLet $k\\geq3$ and consider the pp power ${ G}$ of $\\P_2$ of dimension $k-1$ given by the following formula \n$\\phi(x_1,\\dots x_{k-1},y_1,\\dots,y_{k-1})$ \n\\begin{align*}\n (x_1 = y_2) \\wedge (x_2=y_3) \\wedge \\dots \\wedge (x_{k-2}=y_{k-1}) \\wedge E(x_{k-1},y_1). \n\\end{align*}\nThen ${ G}$ contains the following path of length $k$\n\\[(0,0,\\dots,0)\\to (1,0,\\dots,0) \\to (1,1,\\dots,0) \\to \\dots \\to (1,1,\\dots,1).\\]\nwhich shows that there exists a homomorphism from $\\P_k$ to ${ G}$. Note that whenever there is an edge from $u$ to $v$ in ${ G}$, then the tuple $v$ contains exactly one 1 more than the tuple $u$.\nTherefore, the function $V({ G}) \\to \\{1,\\dots,k\\}$ that maps $v$ to the number of 1's in $v$ is a homomorphism from ${ G}$ to $\\P_k$. Hence $\\P_2\\leq\\P_k$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:maltAndCyclImplyIdempotent}]\nLet ${ G}$ be a finite digraph satisfying $\\Sigma_M$ and $\\Sigma_p$ for every prime $p$. By Lemma~\\ref{lem:maltIffRect} and Theorem~\\ref{thm:maltImpliesPathOrDuoc} there are two cases to consider:\nthe first is that ${ G}$ is homomorphically equivalent to $\\P_k$ for some $k$. Then $\\P_2\\leq { G}$ by Lemma~\\ref{lem:IdempConstructPaths}. \n\nThe second case is that ${ G}$ is homomorphically equivalent to a disjoint union of directed cycles.\nWithout loss of generality we may assume that ${ G}$ is a disjoint union of directed cycles. \nLet $(a_0,\\dots,a_{\\ell-1})$ be a shortest cycle in ${ G}$. Let $p$ be a prime and $k\\in\\mathbb{N}^+\\!$ such that $p\\cdot k=\\ell$, and\nlet $f\\in\\ensuremath{\\operatorname{Pol}}({ G})$ be a function that witnesses that $\\ensuremath{\\operatorname{Pol}}({ G})\\models\\Sigma_p$. \nThen\n\\begin{align*}\n f(a_0,a_k,\\dots,a_{(p-1)\\cdot k})=a=f(a_k,a_{2 k},\\dots,a_{0}).\n\\end{align*}\nSince $f$ is a polymorphism of ${ G}$ there is a directed path of length $k$ from $a$ to $a$. Thus, ${ G}$ contains a directed cycle whose length divides $k$, which contradicts the assumption that $\\ell$ is the length of the shortest directed cycle in ${ G}$. Therefore, $\\ell$ has no prime divisors, and $\\ell=1$. So ${ G}$ contains a loop and hence is homomorphically equivalent to $C_1$; it follows that $\\P_2 \\leq { G}$. \n\\end{proof}\n\n\\section{Proof of the main result}\n\n\nWe use the following general result about when a finite digraph can pp construct a finite disjoint union of cycles.\n\\begin{lemma}[Lemma 6.8 in~\\cite{smooth-digraphs}]\nLet ${C}$ be a finite disjoint union of cycles and let ${ G}$ be a finite digraph. Then \n\\begin{align*}\n{ G}\\leq {C} && \\text{iff} && \\ensuremath{\\operatorname{Pol}}({ G})\\models\\Sigma_{C\\dotdiv c} \\text{ implies } \\ensuremath{\\operatorname{Pol}}({C})\\models\\Sigma_{C\\dotdiv c}\\text{ for all $c\\in\\mathbb{N}^+$\\!.}\n\\end{align*}\n\\end{lemma}\n\nFor the special case that $C=C_p$, there are only two conditions of the form $\\Sigma_{C\\dotdiv c}$, namely $\\Sigma_1$, which is trivial, and $\\Sigma_p$, which is not satisfied by $C_p$. Hence, we obtain the following result. \n\\begin{theorem}\n\\label{thm:notSatSigma}\nIf ${ G}$ is a finite digraph. If $p$ is a prime number such that $\\ensuremath{\\operatorname{Pol}}({ G}) \\not\\models\\Sigma_p$, then ${ G} \\leq{C}_p$.\n\\end{theorem}\n\n\nWe also need a similar results for $\\Sigma_M$ instead of $\\Sigma_p$. \n\n\\begin{lemma}\\label{lem:notSatMalt}\nLet ${ G}$ be a finite digraph. \nIf $\\ensuremath{\\operatorname{Pol}}({ G}) \\not \\models \\Sigma_M$, then ${ G} \\leq {T}_3$. \n\\end{lemma}\n\n\n\\begin{proof\nSince $\\leq$ is transitive we may assume without loss of generality that $\\H = (V,E)$ is a core. By Lemma~\\ref{lem:maltIffRect}, $\\H$ is not totally rectangular. Hence, there are vertices $a,b,c,d \\in V$ such that in ${ G}$ there are directed paths of length $k$ from $a$ to $b$, from $c$ to $b$, from $c$ to $d$, and there is no directed path of length $k$ from $a$ to $d$. Note that by Lemma~\\ref{lem:constants} we are allowed to use constants in pp constructions. \nWe write $x\\stackrel{k}\\to y$\nas a shortcut for the primitive positive formula \n$\\exists u_1,\\dots,u_{k-2} (E(x,u_1) \\wedge E(u_1,u_2) \\wedge \\cdots \\wedge E(u_{k-2},y))$. \nConsider the pp power of ${ G}$ of dimension two given by the formula\n\\begin{align*}\n \\phi(x_1,x_2,y_1,y_2)&\\coloneqq x_1\\stackrel{k}\\to y_2 \\wedge (x_2=d) \\wedge (y_1=a).\n\\end{align*}\n\nLet $\\H$ be the resulting digraph. Consider the vertices $v_0=(c,d)$, $v_1=(a,d)$, and $v_2=(a,b)$ of $\\H$. Note that the only vertex of $\\H$ that can have incoming and outgoing edges is $v_1$. Since there is no path of length $k$ from $a$ to $d$ the vertex $v_1$ does not have a loop. Furthermore, $\\H$ has the edges $(v_0,v_1), (v_1,v_2),$ and $(v_0,v_2)$ (see Figure~\\ref{fig:T3constr}). \nHence, $i\\mapsto v_i$ is an embedding of ${T}_3$ into $\\H$. \nLet $V_0$ be the set of all vertices in $H$ that have outgoing edges and $V_2$ be the set of all vertices in $H$ that have incoming edges. Note that $V_0\\mathbin\\Delta V_2$ consists of $v_1$ and all isolated vertices. Clearly, $V_0\\setminus V_2$, $V_0\\mathbin\\Delta V_2$, and $V_2\\setminus V_0$ form a partition of $V(H)$ and\nthe map\n\\begin{align*}\n v\\mapsto\n \\begin{cases}\n v_2&\\text{if }v\\in V_2\\setminus V_0\\\\\n v_1&\\text{if }v\\in V_0\\mathbin\\Delta V_2\\\\\n v_0&\\text{if }v\\in V_0\\setminus V_2\n \\end{cases}\n\\end{align*}\nis a homomorphism from $\\H$ to ${T}_3$. Hence ${ G} \\leq {T}_3$.\n\\end{proof}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}\n \\node (0) at (0,0) {$(c,d)$};\n \\node (1) at (0,1.5) {$(a,d)$};\n \\node (2) at (0,3) {$(a,b)$};\n \n \\path[->]\n (0) edge (1)\n (1) edge (2)\n (0) edge[bend left=42] (2)\n ;\n \\end{tikzpicture}\n \\caption{The primitive positive construction of $T_3$ in the proof of Lemma~\\ref{lem:notSatMalt}.}\n \\label{fig:T3constr}\n\\end{figure}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:submaximalGraphs}]\nLet ${ G}$ be a digraph such that $\\P_2\\not\\leq { G}$. \nTheorem~\\ref{thm:maltAndCyclImplyIdempotent} implies that either $\\ensuremath{\\operatorname{Pol}}({ G})$ does not satisfy $\\Sigma_M$ or that it does not satisfy $\\Sigma_p$ for some prime $p$. In the first case ${ G} \\leq {T}_3$, by Lemma~\\ref{lem:notSatMalt}. \nIn the second case ${ G}\\leq{C}_p$, by Theorem~\\ref{thm:notSatSigma}.\nHence, all submaximal elements of $\\mathfrak P_{\\Digraphs}$ are contained in \n$\\{{T}_3, {C}_2 ,{C}_3, {C}_5, \\dots\\}$. \nLemma~\\ref{lem:CpConditions1}, Lemma~\\ref{lem:CpConditions2}, and Lemma~\\ref{lem:T3Conditions} imply that these digraphs form an antichain in $\\mathfrak P_{\\Digraphs}$, and hence \neach of these digraphs is submaximal. \n\\end{proof}\n\n\n\nNote that our result implies the following. \n\n\\begin{corollary}\\label{cor:maltAndCyclImplyIdempotent}\nIf a finite digraph ${ G}$ satisfies $\\Sigma_M$, $\\Sigma_2$, $\\Sigma_3$, $\\Sigma_5$, $\\dots$, then any minor condition satisfied by $\\ensuremath{\\operatorname{Pol}}(\\P_2)$ is also satisfied by $\\ensuremath{\\operatorname{Pol}}({ G})$.\n\\end{corollary}\n\nThe statement of Corollary~\\ref{cor:maltAndCyclImplyIdempotent} may also be phrased as\n$$\\{\\Sigma_M, \\Sigma_2, \\Sigma_3, \\Sigma_5, \\dots\\}\\subseteq \\Sigma({ G}) \\quad \\Rightarrow \\quad \\Sigma(\\P_2)\\subseteq \\Sigma({ G}).$$\n\n\\begin{remark}\nWe do not know whether Corollary~\\ref{cor:maltAndCyclImplyIdempotent} holds for arbitrary clones of operations on a finite set, instead of just clones of the form $\\ensuremath{\\operatorname{Pol}}({ G})$ for a finite digraph $G$. However, the statement is false for clones of operations on an infinite set, as illustrated by the clone of operations on ${\\mathbb Q}$ of the form\n$(x_1,\\dots,x_n) \\mapsto a_1x_1+\\cdots+a_nx_n$ for \n$a_1,\\dots,a_n \\in {\\mathbb Q}$ such that $a_1 + \\cdots + a_n = 1$. \nThis clone satisfies $\\Sigma_n$ for every $n \\in \\mathbb N$,\nand contains the function $(x_1,x_2,x_3) \\mapsto x_1 - x_2 + x_3$, so it also satisfies $\\Sigma_M$. \nHowever, it is easy to see that it does not contain an operation $f$ that satisfies\n$$f(x,x,y) = f(y,y,x) = f(x,y,y) = f(y,x,x)$$\nfor all $x,y \\in {\\mathbb Q}$;\nhowever, this minor condition is satisfied by $\\ensuremath{\\operatorname{Pol}}(\\P_2)$ (for example by $f = \\max$). \n\\end{remark}\n\n\\begin{remark}\nMany, but not all the statements that we have shown also apply to \\emph{infinite} digraphs. In Theorem~\\ref{thm:wonderland}, only the forward direction holds if $G$ and $H$ are infinite; however, in this text we only used the forward direction of this theorem. \n\nEvery digraph with a Maltsev polymorphism is totally rectangular even if the digraph\nis infinite. \nThe proof of Theorem~\\ref{thm:maltImpliesPathOrDuoc} of Carvalho, Egri, Jackson, and Niven can be generalised to show that \nevery infinite digraph which is totally rectangular \nhomomorphically equivalent to an infinite disjoint unions of cycles or one of the infinite paths $P^\\infty\\coloneqq(\\mathbb{N},\\{(u,u+1)\\mid u\\in\\mathbb{N}\\})$, \n$P_\\infty\\coloneqq(\\mathbb{N},\\{(u+1,u)\\mid u\\in\\mathbb{N}\\})$, \nthe disjoint union $P_\\infty + P^\\infty$ of $P_\\infty$ and $P^\\infty$, \nand $P^\\infty_\\infty\\coloneqq(\\mathbb{Z},\\{(u,u+1)\\mid u\\in\\mathbb{Z}\\})$. All of these graphs have a Maltsev polymorphism. \n\nInfinite disjoint unions of cycles are clearly not submaximal. \nClearly, $P_2$ cannot pp construct the core digraphs $P_{\\infty}$, $P^{\\infty}$, $P_\\infty + P^\\infty$, and $P^\\infty_\\infty$, \nand these graphs can pp construct $P_2$. Clearly $P_{\\infty}$ and $P^{\\infty}$ pp-construct each other. \nWe do not know whether these graphs are submaximal in the class of all digraphs. \n\\end{remark}\n\n\n\\section{Concluding remarks}\nPrimitive positive constructability orders finite digraphs $H$ by their `strength' with respect to the $H$-coloring problem. Many deep combinatorial statements about graphs and digraphs can be phrased in terms of this order. We showed that at least the top region of the resulting poset can be described completely.\nA full description of the entire poset $\\mathfrak P_{\\Digraphs}$ would be highly desirable.\nWe state three concrete open problems.\n\\begin{enumerate}\n \\item Is $\\mathfrak P_{\\Digraphs}$ a lattice? (Primitive positive constructability is known to form a join meet-lattice on the class of all finite relational structures factored by homomorphic equivalence, but it is not clear to the authors whether the clone product construction for the meet used there can be carried out in the category of digraphs.) \n \\item Does $\\mathfrak P_{\\Digraphs}$ contain infinite ascending chains? (We have seen an infinite antichain in this article; an infinite descending chain of digraphs with a Maltsev polymorphism can be found in~\\cite{smooth-digraphs} and \n the existence of infinite descending chains of digraphs without a Maltsev polymorphism follows from results of~\\cite{BMOOPW}, and also from results in~\\cite{JacksonKN16}.) \n \n \n \n \\item What are the greatest lower bounds of $T_3$ in $\\mathfrak P_{\\Digraphs}$? \n\\end{enumerate}\n\nWe already mentioned that the pp constructability poset on disjoint unions of cycles has been described in~\\cite{smooth-digraphs}; in particular, it contains no infinite ascending chains and is a lattice. \nNote that this result combined with the result of the present paper shows that when exploring $\\mathfrak P_{\\Digraphs}$ it only remains to explore the interval between $K_3$ and $T_3$: \nif a digraph $\\H$ does not have a Maltsev polymorphism, then we proved that it is below $T_3$ (and above $K_3$);\notherwise, it is homomorphically equivalent to a directed path or a disjoint union of cycles and hence falls into the region that has already been completely described. \n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nLet $p$ be an odd prime number and let $\\mathbb{F}_q$ be the finite field with $q=p^M$ elements. Denote by $\\text{W}(\\mathbb{F}_q)$ the ring Witt vectors of $\\mathbb{F}_q$. For each prime number $l$, choose an embedding $\\iota_l:\\bar{\\mathbb{Q}}\\hookrightarrow \\bar{\\mathbb{Q}}_l$. Denote by $\\operatorname{G}_l$ the absolute Galois group $\\operatorname{Gal}(\\bar{\\mathbb{Q}}_l\/\\mathbb{Q}_l)$. Let $S$ be a finite set of prime numbers containing $p$. Denote by $\\mathbb{Q}_S$ the maximal algebraic extension of $\\mathbb{Q}$ which is unramified at each prime $l\\notin S$ and set $\\operatorname{G}_{\\mathbb{Q},S}=\\operatorname{Gal}(\\mathbb{Q}_S\/\\mathbb{Q})$. The weak form of Serre's conjecture states that an odd and irreducible two-dimensional Galois representation \n\\[\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\text{GL}_2(\\mathbb{F}_q)\\]is modular, i.e., lifts to a characteristic zero representation $\\varrho$ attached to an eigencuspform. The strong form asserts that the eigencuspform may be chosen to have optimal level equal to the prime to $p$ part of the Artin conductor of $\\bar{\\varrho}$. Ribet \\cite{ribetLL} proved via a level lowering argument that the weak form implies the strong form. Khare-Wintenberger \\cite{KW2} went on to prove the full statement, building on Ribet's work.\n\\par \nIn \\cite{hamblenramakrishna}, Hamblen and Ramakrishna prove a generalization of the weak form of Serre's conjecture for reducible two-dimensional Galois representations. They impose some conditions on a two-dimensional representation\n$\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\op{GL}_2(\\mathbb{F}_q)$, namely,\n\\begin{enumerate}\n\\item $\\bar{\\varrho}$ is reducible of the form \n\\[\\bar{\\varrho}=\\mtx{\\varphi}{\\ast}{}{1},\\] where $\\varphi:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\mathbb{F}_q^{\\times}$ is a Galois character.\n\\item The representation $\\bar{\\varrho}$ is odd, i.e. if $c\\in \\operatorname{G}_{\\mathbb{Q}}$ denotes complex conjugation, \\[\\det \\bar{\\varrho}(c)=-1.\\]\n\\item The representation $\\bar{\\varrho}$ is indecomposable, i.e. the cohomology class \\[\\ast\\in H^1(\\operatorname{G}_{\\mathbb{Q},S}, \\mathbb{F}_q(\\varphi))\\] is non-zero.\n\\item \nThe Galois character $\\varphi$ is stipulated to satisfy some conditions, for instance, $\\varphi\\neq \\bar{\\chi},\\bar{\\chi}^{-1}$ where $\\bar{\\chi}$ is the mod $p$ cyclotomic character and $\\varphi^2\\neq 1$. Further, the image of $\\varphi$ is stipulated to span $\\mathbb{F}_q$ over $\\mathbb{F}_p$.\n\\item \nThere are further conditions on the restriction $\\bar{\\varrho}_{\\restriction \\operatorname{G}_p}$. The reader may refer to condition 5 of Theorem 2 in \\cite{hamblenramakrishna} for further details.\n\\end{enumerate} \nHamblen and Ramakrishna show that if $\\bar{\\varrho}$ satisfies the above mentioned conditions, then on enlarging the set of ramification $S$ by a finite set of primes $X$, $\\bar{\\varrho}$ has an odd, irreducible, $p$-ordinary lift $\\varrho$ which is unramified outside $S\\cup X$\n \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},S}$};\n \\node (GL2) [right of=GS]{$\\text{GL}_2(\\mathbb{F}_q).$};\n \\node (GL2W) [above of= GL2]{$\\text{GL}_2(\\text{W}(\\mathbb{F}_q))$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\varrho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\varrho$} (GL2W);\n \\end{tikzpicture}\\]This lift is \\textit{geometric} in the sense of Fontaine and Mazur \\cite{fontainemazur}. By the result of Skinner and Wiles in \\cite{skinnerwiles}, the representation $\\varrho$ arises from a $p$-ordinary eigencuspform. This settles the weak form of Serre's conjecture for such $\\bar{\\varrho}$.\n\\par\nThe prospect of generalizing this lifting result leads us to examine higher dimensional Galois representations with image in a smooth group-scheme $\\operatorname{G}$ over $\\text{W}(\\mathbb{F}_q)$. Assume that $\\op{G}_{\\restriction \\mathbb{F}_q}$ is split and reductive and choose a split Borel $\\operatorname{B}_{\/\\mathbb{F}_q}\\subset \\op{G}_{\\restriction \\mathbb{F}_q}$. Let $\\bar{\\rho}$ be a homomorphism \\[\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{G}(\\mathbb{F}_q).\\] Let $\\mathfrak{g}$ denote the Lie-algebra of the adjoint group $\\operatorname{G}_{\\restriction \\mathbb{F}_q}^{ad}$ and $\\Phi(\\operatorname{G}_{\\restriction \\mathbb{F}_q}^{ad})$ be a root system compatible with the choice of Borel. Denote by $\\mathfrak{n}\\subset \\mathfrak{g}$ the span of the positive roots. The $\\mathbb{F}_q$-vector space $\\mathfrak{g}$ acquires an adjoint Galois action\n\\[\\operatorname{Ad}^0\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{Aut}_{\\mathbb{F}_q}(\\mathfrak{g}).\\] Denote by $\\operatorname{Ad}^0\\bar{\\rho}$ the Galois module with underlying vector space $\\mathfrak{g}$. It is imperative that $\\bar{\\rho}$ is \\textit{odd}. For an involution $\\tau\\in \\op{Aut}(\\operatorname{G}_{\\restriction \\mathbb{F}_q})$, let $(\\operatorname{Ad}^0\\bar{\\rho})^{\\tau}$ denote the subspace of $\\operatorname{Ad}^0\\bar{\\rho}$ fixed by $\\tau$. It was shown by E. Cartan that\n\\begin{equation*}\n \\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\tau}\\geq\\dim \\mathfrak{n}\n\\end{equation*}\n(see \\cite[Proposition 2.2]{Yun} for further details).\nThe representation $\\bar{\\rho}$ is \\textit{odd} if equality is achieved for the involution $\\op{ad} \\bar{\\rho}(c)$, i.e.\n\\begin{equation}\\label{oddness}\n \\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}.\n\\end{equation}\nIn particular, the group $\\operatorname{G}$ must contain an element $h=\\bar{\\rho}(c)$ for which equality $\\ref{oddness}$ is achieved. Such an element is said to induce a \\textit{Chevalley involution}. When $n>2$, the general linear group $\\operatorname{GL}_n(\\mathbb{F}_q)$ contains no such element. Hence there are no odd representations for the group $\\operatorname{GL}_n(\\mathbb{F}_q)$ when $n>2$. On the other hand, the general symplectic group $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ does contain elements which induce Chevalley involutions. \n\\par Ramakrishna in \\cite{RamLGR} and \\cite{RamFM} showed that odd, irreducible representations $\\bar{\\rho}:\\op{G}_{\\mathbb{Q}}\\rightarrow \\op{GL}_2(\\bar{\\mathbb{F}}_p)$ satisfying some additional hypotheses exhibit characteristic zero lifts which are \\textit{geometric} in the sense of Fontaine and Mazur. These results provided evidence for Serre's conjecture, before it was proved by Khare and Wintenberger. Taylor in \\cite{taylor} introduced a reformulation of Ramakrishna's method, by showing that the vanishing of a certain \\textit{dual Selmer group} is sufficient in asserting the existence of global Galois deformations with fixed local conditions. This new formulation paved the way for higher dimensional generalizations. In \\cite{partikisthesis}, Patrikis generalized Ramakrishna's lifting theorem to odd representations with \\textit{big image} in $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$. Fakhruddin, Khare and Patrikis studied more general odd, irreducible representations in \\cite{FKP1} and \\cite{FKP2}.\n\\par We assume that our representation has image in $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ for $n\\geq 2$. Associate to a commutative $\\text{W}(\\mathbb{F}_q)$-algebra $R$, a non-degenerate alternating form on $R^{2n}$ prescribed by the matrix\\[J:=\\mtx{}{\\operatorname{Id}_n}{-\\operatorname{Id}_n}{}.\\] The group of general symplectic matrices $\\operatorname{GSp}_{2n}(R)$ consists of matrices $X$ which preserve this form up to a scalar i.e. satisfy\n$X^t J X\\in R^{\\times} \\cdot J$. The similitude character $\\nu:\\operatorname{GSp}_{2n}(R)\\rightarrow R^{\\times}$ is defined by the relation $X^t J X=\\nu(X)\\cdot J$. The space $\\operatorname{Ad}^0\\bar{\\rho}$ is an $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-module with underlying space $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$. The Galois action is prescribed by\n \\[g\\cdot X=\\bar{\\rho}(g) X \\bar{\\rho}(g)^{-1}\\]where $g\\in \\operatorname{G}_{\\mathbb{Q},S}$ and $X\\in \\operatorname{sp}_{2n}(\\mathbb{F}_q)$. Let $\\operatorname{B}(R)$ be the Borel subgroup consisting of matrices\n\\[M=\\mtx{C}{CD}{}{\\xi (C^t)^{-1}}\\]where $C\\in \\operatorname{GL}_n(R)$ is upper triangular, $D\\in \\operatorname{GL}_n(R)$ is symmetric and $\\xi\\in R^{\\times}$. Note that in this setting, $\\op{B}$ is defined over $\\text{W}(\\mathbb{F}_q)$. Denote by $U_1\\subset \\op{B}$ the unipotent subgroup.\n\\par\nLet $\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ be a continuous Galois representation with image in $\\op{B}(\\mathbb{F}_q)$. Composing $\\bar{\\rho}$ with the similitude-character $\\bar{\\nu}$ defines a Galois character denoted by $\\bar{\\kappa}$. Denote by $\\mathcal{T}\\subseteq \\operatorname{GSp}_{2n}$ the diagonal torus and $e_{i,j}\\in \\op{GL}_{2n}(\\mathbb{F}_q)$ the matrix with $1$ in the $(i,j)$-position and $0$ in all other positions. Set $\\mathfrak{t}$ for the $\\mathbb{F}_q$-span of $H_1,\\dots, H_n$, where $H_i:=e_{i,i}-e_{n+i,n+i}$. Let $L_1, \\dots, L_n\\in \\mathfrak{t}^*$ be the dual basis of $H_1,\\dots, H_n$. An integer linear combination $\\lambda$ of $L_1, \\dots, L_n$ is viewed as character on the torus $\\mathcal{T}(\\mathbb{F}_q)$, which is trivial on the center of $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$. Via the natural quotient map $\\op{B}\\rightarrow \\mathcal{T}$, a character on $\\mathcal{T}$ induces a character on $\\op{B}$. The character on $\\op{B}$ induced by $\\lambda$ is denoted by \\[\\omega_{\\lambda}:\\op{B}(\\mathbb{F}_q)\\rightarrow \\mathbb{F}_q^{\\times}.\\] Associated to $\\omega_{\\lambda}$ is the Galois character\n\\[\\sigma_{\\lambda}=\\omega_{\\lambda}\\circ \\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\mathbb{F}_q^{\\times}.\\] Let \"$1$\" be a formal symbol for the trivial linear combination of $L_1,\\dots, L_n$ and set $\\sigma_1$ equal to the trivial character. The roots $\\Phi=\\Phi(\\operatorname{Ad}^0\\bar{\\rho}, \\mathfrak{t})$ are specified by\n\\[\\begin{split}\n\\Phi=& \\{\\pm 2L_1, \\dots, \\pm 2 L_n\\} \\\\\\cup &\\{\\pm(L_i+L_j)\\mid 1\\leq i2n$,\n\\item\\label{thc2}$\\bar{\\rho}$ is odd, i.e.\n$\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}$.\n\\item\\label{thc3}The image of $\\bar{\\rho}$ contains the unipotent subgroup $\\operatorname{U}_1(\\mathbb{F}_q)$.\n\\item\\label{thc4}Both the following conditions on the distinctness of the characters $\\{\\sigma_{\\lambda}\\}$ are satisfied:\n\\begin{enumerate}\n \\item For $\\lambda, \\lambda'\\in \\Phi\\cup \\{1\\}$ such that $\\lambda\\neq \\lambda'$, $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$-twist of $\\sigma_{\\lambda'}$.\n \\item Moreover for $\\lambda, \\lambda'\\in \\Phi\\cup \\{1\\}$ not necessarily distinct, $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$-twist of $\\bar{\\chi}\\sigma_{\\lambda'}$.\n\\end{enumerate}\n\\item\\label{thc5}\nFor each of the roots $\\lambda\\in \\Phi$, the $\\mathbb{F}_p$-linear span of the image of $\\sigma_{\\lambda}$ in $\\mathbb{F}_q$ is $\\mathbb{F}_q$.\n\n\\item\\label{thc7}\nAt each prime $v\\in S$ such that $v\\neq p$, there is a liftable local deformation condition $\\mathcal{C}_v$ with tangent space $\\mathcal{N}_v$ of dimension \n\\[\\dim \\mathcal{N}_v=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\]\\item\\label{thc8}Tilouine's regularity conditions $(\\operatorname{REG})$ and $(\\operatorname{REG})^*$ are satisfied, i.e. \n\\[H^0(\\op{G}_p, \\operatorname{Ad}^0\\bar{\\rho}\/\\mathfrak{b})=0\\text{ and }H^0(\\op{G}_p, (\\operatorname{Ad}^0\\bar{\\rho}\/\\mathfrak{b})(\\bar{\\chi}))=0.\\]\n\\end{enumerate} Let $\\kappa$ be a fixed choice of a lift of the character $\\bar{\\kappa}$ such that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer divisible by $p(p-1)$ and $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$. Then $\\exists$ a finite set of auxiliary primes $X$ disjoint from $S$ and a lift $\\rho$ \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},S}$};\n \\node (GL2) [right of=GS]{$\\operatorname{GSp}_{2n}(\\mathbb{F}_q).$};\n \\node (GL2W) [above of= GL2]{$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q))$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\rho$} (GL2W);\n \\end{tikzpicture}\\] for which \n\\begin{enumerate}\n\\item $\\rho$ is irreducible,\n\\item $\\rho$ is $p$-ordinary (in the sense of \\cite[section 4.1]{patrikisexceptional}),\n\\item $\\nu\\circ \\rho= \\kappa$,\n\\item for $v\\in S\\backslash \\{p\\}$, the restriction to the decomposition group $\\rho_{\\restriction \\operatorname{G}_v}\\in \\mathcal{C}_v$.\n\\end{enumerate}\n\\end{Th}\n\nThe lift $\\rho$ is geometric in the sense of Fontaine and Mazur. For $\\lambda\\in \\Phi$, setting $\\lambda=-\\lambda'$ in condition $\\eqref{thc4}$, we have that $\\sigma_{\\lambda}^2\\neq 1$. Note that the conditions also imply that $\\sigma_{\\lambda}\\neq \\bar{\\chi},\\bar{\\chi}^{-1}$. This is reminiscent of the condition $\\varphi^2\\neq 1$ of Hamblen-Ramakrishna. In particular, condition $\\eqref{thc4}$ implies that $p>2$. The requirement $p>2n$ is primarily made so that we may suitably work with the exponential map in various places.\n\\par It is a consequence of Tilouine's regularity conditions that the ordinary deformations of $\\bar{\\rho}_{\\restriction \\op{G}_p}$ constitute a liftable deformation condition $\\mathcal{C}_p$ for which the tangent space $\\mathcal{N}_p$ has dimension:\n\\[\\dim \\mathcal{N}_p=h^0(\\op{G}_p,\\operatorname{Ad}^0\\bar{\\rho})+\\dim \\mathfrak{n}.\\] For a discussion on the ordinary deformation condition and Tilioune's regularity conditions, the reader may refer to \\cite[section 4]{patrikisexceptional}. With reference to condition $\\eqref{thc7}$, the reader may consult \\cite[sections 4.3 and 4.4]{patrikisexceptional} for examples of such deformation conditions.\n\\par When $n=1$, the unipotent group is abelian and examples of such Galois representations $\\bar{\\rho}$ are constructed via class field theory. The reader may, for instance, refer to \\cite{ribetclassgroups}, \\cite[section 7]{hamblenramakrishna} and \\cite{ray} for more details. In section $\\ref{examples}$, examples of Galois representations $\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)$ satisfying the conditions of Theorem $\\ref{main}$ are constructed. It is shown that if $p\\geq 23$ is a regular prime, there exists a Galois representation\n \\[\\bar{\\rho}=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 &\\ast & \\ast & \\ast \\\\\n & 1 & \\ast & \\ast \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & \\ast & \\bar{\\chi}^9\n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)\\] which satisfies the conditions of Theorem $\\ref{main}$.\n\\subsection*{Acknowledgements}\nThe author is very grateful to his advisor Ravi Ramakrishna for introducing him to the fascinating subject of Galois deformations. He thanks Brian Hwang and Nicolas Templier for fruitful conversations. The author also appreciates the suggestions made by the anonymous referee which have led to significant improvement of the article.\n\\section{Notation}\\label{notationsection}\n\\begin{itemize}\n\\item For an $\\mathbb{F}_q$-vector space $M$, set $\\dim M:=\\dim_{\\mathbb{F}_q} M$.\n\\item At every prime $v$, choose an embedding $\\iota_v:\\bar{\\mathbb{Q}}\\hookrightarrow\\bar{\\mathbb{Q}}_v$. The absolute Galois group $\\operatorname{G}_v=\\operatorname{Gal}(\\bar{\\mathbb{Q}}_v\/\\mathbb{Q}_v)$ is identified with the decomposition group of the prime dividing $v$ determined by $\\iota_v$. \n\\item \n Let $e_{i,j}$ denote the $2n\\times 2n$ square matrix with coefficients in $\\mathbb{F}_q$ with $1$ in the $(i,j)$-th position and $0$ at all other positions. \\item The space $\\operatorname{Ad}^0\\bar{\\rho}$ is an $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-module with underlying space $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$. The Galois action is prescribed by\n \\[g\\cdot X=\\bar{\\rho}(g) X \\bar{\\rho}(g)^{-1}\\]where $g\\in \\operatorname{G}_{\\mathbb{Q},S}$ and $X\\in \\operatorname{sp}_{2n}(\\mathbb{F}_q)$.\n \\item The space of diagonal matrices in $\\operatorname{Ad}^0\\bar{\\rho}$ is denoted by $\\mathfrak{t}$. Let $H_1,\\dots, H_n$ be the basis of $\\mathfrak{t}$ defined by $H_i:=e_{i,i}-e_{n+i,n+i}$. Let $L_1, \\dots, L_n\\in \\mathfrak{t}^*$ be the dual basis.\n \\item Let $\\Phi$ be the set of roots of $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$ and $\\lambda_1,\\dots, \\lambda_n\\in \\Phi$ be the simple roots defined as follows\n \\[\\lambda_i:=\\begin{cases}\n L_i-L_{i+1}\\text{ for }i2n$. The group $U_1$ is the unipotent subgroup of $\\op{B}(\\mathbb{F}_q)$.\n\\item Throughout, $h^i$ will be an abbreviation for $\\dim H^i$. For instance, $h^i(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ is an abbreviation for $\\dim H^i(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$.\n\\item Let $M$ be an $\\mathbb{F}_q[\\op{G}_S]$-module, let $\\Sh^i_S(M)$ consist of cohomology classes $f\\in H^i(\\operatorname{G}_{\\mathbb{Q},S}, M)$ such that $f_{\\restriction \\op{G}_v}=0$ for all $v\\in S$.\n\\end{itemize}\n\\section{The General Lifting Strategy}\\label{section3}\n\\par Let $\\bar{\\varrho}$ be a Galois representation $\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{GL}_2(\\bar{\\mathbb{F}}_p)$ which is irreducible, odd and unramified outside finitely many primes. Ramakrishna in \\cite{RamFM} and \\cite{RamLGR} showed that if $\\bar{\\varrho}$ satisfies additional conditions, it lifts to a continuous Galois representation $\\varrho$ which is geometric in the sense of Fontaine and Mazur. In other words, $\\varrho$ is odd, unramified outside finitely many primes and $\\varrho_{\\restriction \\operatorname{G}_p}$ is de Rham. This geometric lifting theorem provided evidence for the weak version of Serre's conjecture before it was proved by Khare and Wintenberger. The geometric lifting construction was adapted to the reducible setting in \\cite{hamblenramakrishna}. The main result of this manuscript is a higher dimensional generalization of the lifting theorem of Hamblen-Ramakrishna. The basic strategy involves successively lifting $\\bar{\\rho}$ to a characteristic zero irreducible geometric representation $\\rho$ by successively lifting $\\rho_m$ to $\\rho_{m+1}$\n\\[\\begin{tikzpicture}[node distance = 2.0 cm, auto]\n \\node (GSX) at (0,0){$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GL2) at (5,0){$\\operatorname{GSp}_{2n}(\\mathbb{F}_q).$};\n \\node (GL2Wn) at (3,2)[above of= GL2]{$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^{m})$};\n \\node (GL2Wnplus1) at (5,4){$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^{m+1})$};\n \\draw[->] (GSX) to node [swap]{$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2Wn) to node {} (GL2);\n \\draw[->] (GSX) to node [swap]{$\\rho_m$} (GL2Wn);\n \\draw[->] (GL2Wnplus1) to node {} (GL2Wn);\n \\draw[dashed,->] (GSX) to node {$\\rho_{m+1}$} (GL2Wnplus1);\n \\end{tikzpicture}\\] \n \n \\begin{Def} Let $\\mathcal{C}$ be the category of coefficient rings over $\\text{W}(\\mathbb{F}_q)$ with residue field $\\mathbb{F}_q$. The objects of this category consist of local $\\text{W}(\\mathbb{F}_q)$-algebras $(R,\\mathfrak{m})$ for which\n \\begin{itemize}\n \\item $R$ is complete and Noetherian,\n \\item $R\/\\mathfrak{m}$ is isomorphic to $\\mathbb{F}_q$ as a $\\text{W}(\\mathbb{F}_q)$-algebra. The residual map \\[\\phi:R\\rightarrow \\mathbb{F}_q\\]is the composite of the quotient map $R\\rightarrow R\/\\mathfrak{m}$ with the unique isomorphism of $W(\\mathbb{F}_q)$-algebras $R\/\\mathfrak{m}\\xrightarrow{\\sim}\\mathbb{F}_q$.\n \\end{itemize} A morphism $F:(R_1,\\mathfrak{m}_1)\\rightarrow (R_2,\\mathfrak{m}_2)$ is a homorphism of local rings which is also a $\\text{W}(\\mathbb{F}_q)$-algebra homorphism. Recall that $\\kappa$ is a fixed choice of lift of $\\bar{\\kappa}$. Let $\\kappa_v$ denote the restriction of $\\kappa$ to $\\operatorname{G}_v$.\n \\end{Def}\n \\par Let $v$ be a prime and $R\\in \\mathcal{C}$. Denote by $\\phi^*: \\operatorname{GSp}_{2n}(R)\\rightarrow \\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ the group homomorphism induced by the residual homomorphism $\\phi: R\\rightarrow \\mathbb{F}_q$. We say that $\\rho_R:\\operatorname{G}_v\\rightarrow \\operatorname{GSp}_{2n}(R)$ is an $R$-lift of $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ if $\\phi^*\\circ \\rho_R=\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$, i.e. the following diagram commutes\n \\[ \\begin{tikzpicture}[node distance = 2.2 cm, auto]\n \\node(G) at (0,0){$\\operatorname{G}_{v}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$.};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(R)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$} (A);\n \\draw[->] (B) to node{$\\phi^*$} (A);\n \\draw[->] (G) to node {$\\rho_R$} (B);\n \\end{tikzpicture}\\]Further, we shall require that the similitude character of $\\rho_R$ coincides with the composite of $\\kappa_v$ with the homomorphism $W(\\mathbb{F}_q)^{\\times}\\rightarrow R^{\\times}$ induced by the structure map.\n \\par Two lifts $\\rho_R$ and $\\rho_R'$ are said to be strictly-equivalent if there is\n \\[A\\in \\text{ker}\\lbrace \\operatorname{GSp}_{2n}(R)\\xrightarrow{\\phi^*} \\operatorname{GSp}_{2n}(\\mathbb{F}_q)\\rbrace \\]\n such that \n $\\rho_R=A\\rho_R' A^{-1}$. A deformation is a strict equivalence class of lifts. Let $\\operatorname{Def}_v(R)$ be the set of $R$-deformations of $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$. The association $R\\mapsto \\operatorname{Def}_v(R)$ defines a covariant functor \\[\\operatorname{Def}_v:\\mathcal{C}\\rightarrow \\operatorname{Sets}. \\]\n The tangent space $\\operatorname{Def}_v(\\mathbb{F}_q[\\epsilon]\/(\\epsilon^2))$ naturally acquires the structure of an $\\mathbb{F}_q$-vector space and is isomorphic to $H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})$. Under this association, a cohomology class $f$ is identified with the deformation $(\\operatorname{Id}+\\epsilon f)\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$.\n \\par For $m\\in \\mathbb{Z}_{\\geq 2}$, the deformations $\\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$ are equipped with action of the cohomology group $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$. For $\\varrho_m\\in \\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$ and $f\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$, the twist of $\\varrho_m$ by $f$ is defined by the formula $(\\op{Id}+p^{m-1}f)\\varrho_m$. The set of deformations $\\varrho_m$ of a fixed $\\varrho_{m-1}\\in \\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m-1})$ is either empty or in bijection with $H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})$.\n \\begin{Def}\\label{defconditiondef} (see \\cite{taylor}) We say that a sub-functor $\\mathcal{C}_v$ of $\\operatorname{Def}_v$ is a deformation condition if (1) to (3) below are satisfied. If condition (4) is satisfied, $\\mathcal{C}_v$ is said to be liftable.\n \\begin{enumerate}\n \\item First, we require that $\\mathcal{C}_v(\\mathbb{F}_q)=\\{\\bar{\\rho}_{\\restriction \\operatorname{G}_v}\\}.$\n \\item For $R_1$ and $R_2$ be $\\mathcal{C}$, let $\\rho_1\\in \\mathcal{C}_v(R_1)$ and $\\rho_2\\in \\mathcal{C}_v(R_2)$. Let $I_1$ be an ideal in $R_1$ and $I_2$ an ideal in $R_2$ such that there is an isomorphism $\\alpha:R_1\/I_1\\xrightarrow{\\sim} R_2\/I_2$ satisfying \\[\\alpha(\\rho_1 \\;\\text{mod}{I_1})=\\rho_2 \\;\\text{mod}{I_2}.\\] Let $R_3$ be the fibred product \\[R_3=\\lbrace(r_1,r_2)\\mid \\alpha(r_1\\text{mod} I_1)=r_2 \\text{mod} I_2\\rbrace\\] and $\\rho_3$ the $R_3$-deformation induced from $\\rho_1$ and $\\rho_2$. Then $\\rho_3$ satisfies $\\mathcal{C}_v(R_3)$.\n \\item Let $R\\in \\mathcal{C}$ with maximal ideal $\\mathfrak{m}_R$. If $\\rho\\in \\operatorname{Def}_v(R)$ is such that $\\rho\\mod{\\mathfrak{m}_R^n}$ satisfies $\\mathcal{C}_v$ for all $n\\in \\mathbb{Z}_{\\geq 1}$, then $\\rho$ also satisfies $\\mathcal{C}_v$.\n \\item Let $R\\in \\mathcal{C}$ and $I$ an ideal such that $I.\\mathfrak{m}_R=0$. For $\\rho\\in \\mathcal{C}_v(R\/I)$, there exists $\\tilde{\\rho}\\in \\mathcal{C}_v(R)$ such that $\\rho=\\tilde{\\rho}\\mod{I}$.\n \\end{enumerate}\n \\end{Def}\n Let $\\mathcal{C}_v$ be a local deformation condition at the prime $v$. The tangent space $\\mathcal{N}_v$ consists of $f\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$, such that $(\\op{Id}+\\epsilon f) \\bar{\\rho}_{\\restriction \\op{G}_v}\\in \\mathcal{C}_v(\\mathbb{F}_q[\\epsilon]\/(\\epsilon^2))$. The action of $\\mathcal{N}_v$ on $\\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$ stabilizes $\\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$. In other words, if $\\varrho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$ and $f\\in \\mathcal{N}_v$, then \n \\[(\\op{Id}+p^{m-1}f)\\varrho_{m}\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m). \\]It is assumed that each prime $v\\in S\\backslash \\{p\\}$ is equipped with a liftable local deformation condition $\\mathcal{C}_v$ such that \n\\[\\dim \\mathcal{N}_v=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\] The reader may consult \\cite[sections 4.3 and 4.4]{patrikisexceptional} for examples of such deformation conditions. The deformation condition $\\mathcal{C}_p$ is the ordinary deformation condition. Since we have assumed that Tilouine's regularity conditions are satisfied (cf. \\cite[section 4.1]{patrikisexceptional}), $\\mathcal{C}_p$ is liftable and the tangent space $\\mathcal{N}_p$ has dimension equal to\n\\[\\dim \\mathcal{N}_p=h^0(\\operatorname{G}_p, \\operatorname{Ad}^0\\bar{\\rho})+\\dim \\mathfrak{n},\\]see \\cite[Proposition 4.4]{patrikisexceptional}. We allow the successive lifts $\\rho_m$ to be ramified at a set of primes $S\\cup X$. Each auxiliary prime $v\\in X$ is equipped with a liftable subfunctor $\\mathcal{C}_v$ of $\\operatorname{Def}_v$. These auxiliary primes are referred to as trivial primes and were introduced by Hamblen and Ramakrishna in the two-dimensional setting \\cite[section 4]{hamblenramakrishna}. These are primes $v\\equiv 1\\mod{p}$, not contained in $S$, at which $\\bar{\\rho}_{\\restriction \\op{G}_v}$ the trivial representation and $v\\not\\equiv 1\\mod{p^2}$. We use a higher dimensional generalization of the deformation functor $\\mathcal{C}_v$ at a trivial prime $v$, due to Fakhruddin, Khare and Patrikis \\cite[Definition 3.1]{FKP1}. At each trivial prime $v$ there is a subspace $\\mathcal{N}_v$ of $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ of dimension $h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ which behaves like a versal tangent space. For $m\\geq 3$, the action of $\\mathcal{N}_v$ on $\\operatorname{Def}(\\text{W}(\\mathbb{F}_q)\/p^m)$ stabilizes $\\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$. This is proved in the $\\op{GL}_2$-case by Hamblen-Ramakrishna, see \\cite[Corollory 25, 29]{hamblenramakrishna}. For more general groups, we refer to Fakhruddin-Khare-Patrikis \\cite[Lemma 3.6,3.10]{FKP1} for the precise statement. However, this is not the case for $m=2$.\n\\par Let $X$ be a finite set of trivial primes disjoint from $S$. For $v\\in S\\cup X$, set $\\mathcal{N}_v^{\\perp}\\subseteq H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho}^*)$ to be the orthogonal complement of $\\mathcal{N}_v$ with respect to the non-degenerate Tate pairing \n\\[H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\times H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow H^2(\\operatorname{G}_v, \\mathbb{F}_q(\\bar{\\chi}))\\xrightarrow{\\sim}\\mathbb{F}_q.\\] Set $\\mathcal{N}_{\\infty}=0$ and $\\mathcal{N}_{\\infty}^{\\perp}=0$. The Selmer-condition $\\mathcal{N}$ is the tuple $\\{\\mathcal{N}_v\\}_{v\\in S\\cup X\\cup \\{\\infty\\}}$ and the dual Selmer condition $\\mathcal{N}^{\\perp}$ is $\\{\\mathcal{N}_v^{\\perp}\\}_{v\\in S\\cup X\\cup\\{\\infty\\}}$. Attached to $\\mathcal{N}$ and $\\mathcal{N}^{\\perp}$ are the Selmer and dual-Selmer groups defined as follows:\n \\[H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}):=\\text{ker}\\left\\{ H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})\\xrightarrow{\\operatorname{res}_{S\\cup X}} \\bigoplus_{v\\in S\\cup X} \\frac{H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\right\\}\\]\n and\n \\[H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^*):=\\text{ker}\\left\\{ H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*})\\xrightarrow{\\op{res}_{S\\cup X}'} \\bigoplus_{v\\in S\\cup X} \\frac{H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)}{\\mathcal{N}_v^{\\perp}}\\right\\}\\]\n respectively. The following formula is due to Wiles (see \\cite[Theorem 8.7.9]{NW}):\n \\begin{equation}\\label{wilesformula}\\begin{split}h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})-h^1_{\\mathcal{N}^{\\perp}}(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*})&=h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\ &+\\sum_{v\\in S\\cup X\\cup \\{\\infty\\}} \\left(\\dim \\mathcal{N}_v-h^0(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\right).\\\\\\end{split}\\end{equation} Since $\\bar{\\rho}$ is odd, one has that $h^0(\\op{G}_{\\infty}, \\operatorname{Ad}^0\\bar{\\rho})=\\dim \\mathfrak{n}$. It follows from the above formula that the dimensions of the Selmer group and dual Selmer group coincide, i.e.\n \\[h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})=h^1_{\\mathcal{N}^{\\perp}}(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*}).\\]The Selmer and dual Selmer groups fit into a long exact sequence called the Poitou-Tate sequence. We only point out that the cokernel of $\\op{res}_{S\\cup X}$ injects into $H^1_{\\mathcal{N}^{\\perp}}(\\op{S}_{S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}$. In particular, if the Selmer group is zero, then so is the dual Selmer group, in which case the restriction map $\\operatorname{res}_{S\\cup X}$ is an isomorphism. Since the spaces $\\mathcal{N}_v$ at a trivial prime $v$ stabilize lifts only past mod $p^3$, it becomes necessary to produce a mod $p^3$ lift $\\rho_3$ of $\\bar{\\rho}$ before applying the general lifting-strategy. All deformations $\\rho_m$ discussed in this paper will have similitude character equal to $\\kappa\\mod{p^m}$.\n \\par The three main steps are as follows:\n \\begin{enumerate}\n \\item first it is shown that there is a finite set of trivial primes $X_1$ disjoint from $S$ such that the representation $\\bar{\\rho}$ lifts to a mod $p^2$ representation $\\rho_2$ which is unramified outside $S\\cup X_1$.\n \\item It is shown in \\cite[section 5]{hamblenramakrishna} that there is a finite set of trivial primes $X_2\\supset X_1$ disjoint from $S$ and a mod $p^3$ lift $\\rho_3$ of $\\rho_2$ which satisfies the following conditions\n \\begin{itemize}\n \\item $\\rho_3$ is irreducible, i.e. does not contain a free rank one Galois stable $\\text{W}(\\mathbb{F}_q)\/p^3$-submodule.\n \\item It is unramified outside $S\\cup X_2$.\n \\item The lift $\\rho_3$ is also arranged to satisfy conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X_2$.\n \\end{itemize}\n This strategy for getting past mod-$p^2$ is based on the methods developed by Khare, Larsen and Ramakrishna in \\cite{KLR}.\n \\item At this stage, all that remains to be shown is that the set of primes $X_2$ may be further enlarged to a set of trivial primes $X$ which is disjoint from $S$ such that the Selmer group $H^1_{\\mathcal{N}}(\\operatorname{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})$ is equal to zero.\n \\end{enumerate}\n \\par The rest of the argument warrants some explanation. Since the Selmer group is zero, the map $\\op{res}_{S\\cup X}$ is an isomorphism. Suppose for $m\\geq 3$ and $\\rho_m$ is a mod $p^m$ lift of $\\rho_3$ which is unramified outside $S\\cup X$ and satisfies the conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X$. We show that $\\rho_m$ may be lifted to $\\rho_{m+1}$ which satisfies the same conditions. Since the dual Selmer group is zero, so is $\\Sh^1_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho}^*)$, and it follows from global-duality that $\\Sh^2_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho})$ is zero. Since local condition $\\mathcal{C}_v$ is liftable, there are no local obstructions to lifting ${\\rho_m}$. The cohomological obstruction to lifting $\\rho_{m}$ to $\\rho_{m+1}$ is a class in $\\Sh^2_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho})$ and hence is zero. As a result, $\\rho_m$ does lift one more step to $\\rho_{m+1}$. In order to complete the inductive argument, it is shown that $\\rho_{m+1}$ can be chosen to satisfy the conditions $\\mathcal{C}_v$. After picking a suitable global cohomology class $z\\in H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})$ and replacing $\\rho_{m+1}$ by its twist $(\\operatorname{Id}+p^{m}z)\\rho_{m+1}$, this may be arranged. At each prime $v\\in S\\cup X$, there is a cohomology class $z_v\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that the twist $(\\operatorname{Id}+p^mz_v){\\rho_{m+1}}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$. Since we assume that $m\\geq 3$, we have that $\\mathcal{N}_v$ stabilizes $\\mathcal{C}_v$. For $v\\in S\\cup X$, the elements $z_v$ are defined modulo $\\mathcal{N}_v$. Since $\\operatorname{res}_{S\\cup X}$ is an isomorphism, the tuple\n $(z_v)\\in \\bigoplus_{v\\in S\\cup X} H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\/\\mathcal{N}_v$ arises from a unique global cohomology class $z$ which is unramified outside $S\\cup X$. After replacing $\\rho_{m+1}$ by $(\\op{Id}+p^m z) \\rho_{m+1}$, it satisfies the conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X$. This completes the inductive lifting argument.\n\\section{Preliminaries}\n\\par In this section, we prove a number of Galois theoretic results which will be applied in later sections. Let $M$ be a finite abelian group with $\\op{G}_{\\mathbb{Q}}$-action and $E$ be a number field. Denote by $E(M)$ the extension of $E$ \\textit{cut out} by $M$. In other words, it is the Galois extension of $E$ which is fixed by the kernel of the action of $\\op{G}_{E}$ on $M$. Let $M_1,\\dots, M_k$ be finite abelian groups on which $\\op{G}_{\\mathbb{Q}}$ acts. Denote by $E(M_1,\\dots, M_k)$ the composite of the fields $E(M_1)\\cdots E(M_k)$. Let $K:=\\mathbb{Q}(\\bar{\\rho}, \\mu_p)$ and $L:=\\mathbb{Q}(\\bar{\\rho})$ and set $\\op{G}':=\\op{Gal}(K\/\\mathbb{Q})$ and $\\op{G}:=\\op{Gal}(L\/\\mathbb{Q})$. Let $F$ be the subfield $\\mathbb{Q}(\\varphi_1,\\dots, \\varphi_n, \\bar{\\kappa})$ of $L$, where we recall that the characters $\\varphi_1,\\dots, \\varphi_n$ are as in $\\eqref{introducingbarrho}$. Denote by $N':=\\operatorname{Gal}(K\/F(\\mu_p))$ and $N:=\\operatorname{Gal}(L\/F)$. The groups $\\op{G},\\op{G}', N$ and $N'$ are depicted in the following field diagram\n\\begin{equation*}\n\\begin{tikzpicture}[node distance = 1.5cm, auto]\n \\node(Q) {$\\mathbb{Q}.$};\n \\node (L) [above of =Q]{$F$};\n \\node (E) [above of=L, right of=L] {$F(\\mu_p)$};\n \\node (F) [above of=L, left of =L] {$\\mathbb{Q}(\\bar{\\rho})$};\n \\node (K) [above of=E, left of=E] {$\\mathbb{Q}(\\bar{\\rho},\\mu_p)$};\n \\draw[-] (Q) to node {} (L);\n \\draw[-] (L) to node {} (E);\n \\draw[-] (L) to node {\\scriptsize$N$} (F);\n \\draw[-] (F) to node {} (K);\n \\draw[-] (E) to node [swap]{\\scriptsize$N'$} (K);\n \\end{tikzpicture}\n \\end{equation*}\n Condition $\\eqref{thc3}$ of Theorem $\\ref{main}$ asserts that the image of $\\bar{\\rho}$ contains $U_1(\\mathbb{F}_q)$. Therefore $N$ may be identified with $\\bar{\\rho}(N)=U_1(\\mathbb{F}_q)$. In particular the abelianization $N^{ab}$ may be identified with $U_1(\\mathbb{F}_q)\/U_2(\\mathbb{F}_q)$. Since $N$ is a $p$-group and $[F(\\mu_p):F]$ is coprime to $p$, it follows that $\\mathbb{Q}(\\bar{\\rho})$ and $F(\\mu_p)$ are linearly disjoint over $F$. It follows that $N$ is canonically isomorphic to $N'$. The inclusion of $\\mathcal{T}$ into $\\op{B}$ is a section of the quotient map $\\op{B}\\rightarrow \\mathcal{T}$. This induces a semi-direct product decomposition $\\op{B}=U_1\\rtimes \\mathcal{T}$. Let $\\mathcal{T}'$ be the intersection of the image of $\\bar{\\rho}$ with $\\mathcal{T}$. The group $\\op{G}$ may be identified with the image of $\\bar{\\rho}$. It is easy to see that $\\op{G}$ has a semi-direct product decomposition $\\op{G}\\simeq \\bar{\\rho}(\\op{G})=U_1(\\mathbb{F}_q)\\rtimes \\mathcal{T}'$.\\begin{Lemma}\\label{l1}\nSuppose $0<|k|\\leq 2n-1$, there is an isomorphism of $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-modules\n\\[(\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1}\\simeq \\bigoplus_{ht \\lambda=k} \\mathbb{F}_q(\\sigma_{\\lambda}).\\]On the other hand, \\[(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=\\mathfrak{b}\/\\mathfrak{n}\\simeq \\mathfrak{t}.\\]\n\\end{Lemma}\n\\begin{proof}\nLet $\\lambda$ be of height $k$. Let $X\\in (\\operatorname{Ad}^0\\bar{\\rho})_{\\lambda}$, we observe that\n\\[\\bar{\\rho}(g)\\cdot X \\cdot \\bar{\\rho}(g)^{-1}\\equiv \\sigma_{\\lambda}(g) X \\mod{(\\operatorname{Ad}^0\\bar{\\rho})_{k+1}}.\\] Likewise, for $X\\in \\mathfrak{b}$, the conjugation action on $X$ modulo $\\mathfrak{n}$ is trivial.\n\\end{proof}\n\n\\par Let $\\zeta$ be a non-zero element of $\\mathbb{F}_q(\\bar{\\chi})$. For $i=1,\\dots, n$, set $\\delta_{i,j}=\\zeta$ if $i=j$ and $0$ otherwise. Likewise, for roots $\\lambda$ and $\\gamma$, set $\\delta_{\\lambda, \\gamma}$ to equal $\\zeta$ if $\\lambda=\\gamma$ and $0$ otherwise. Denote by $X_{\\lambda}^*$ and $H_i^*$ the elements of $\\operatorname{Ad}^0\\bar{\\rho}^*$ which are defined by the following relations: \\begin{equation}\\label{XHdual}\\begin{split}& X_{\\lambda}^*(X_{\\gamma})=\\delta_{\\lambda,\\gamma}\\text{ and }X_{\\lambda}^*(H_i)=0,\\\\\n & H_i^*(X_{\\lambda})=0 \\text{ and }H_i^*(H_j)=\\delta_{i,j}.\\end{split}\\end{equation} The element $H_i^*\\in \\operatorname{Ad}^0\\bar{\\rho}^*$ should not be confused with $L_i\\in \\mathfrak{t}^*$. Let $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}}$ be the span of $H_1^*,\\dots, H_n^*$ and $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$ the span of $X_{-\\lambda}^*$. Let $P$ be a Galois-stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Associated to $P$ are its eigenspaces for the action of $\\bar{\\rho}^{-1}(\\mathcal{T})$. For $\\lambda\\in \\Phi\\cup \\{1\\}$, set $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ to be the intersection of $P$ with $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$. Likewise, associate to a Galois stable subgroup $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$, an eigenspace $Q_{\\sigma_{\\lambda}}$. Define $Q_1$ to be the intersection $Q\\cap \\mathfrak{t}$. For $\\lambda \\in \\Phi$, denote by $Q_{\\sigma_{\\lambda}}$ the intersection $Q\\cap (\\operatorname{Ad}^0\\bar{\\rho})_{\\lambda}$.\n \\par The representation $\\bar{\\rho}$ factors through $\\op{G}$. Let $\\mathbb{T}$ be the subgroup of $\\op{G}'$ consisting of $g$ such that $\\bar{\\rho}(g)\\in \\mathcal{T}$. For $\\lambda\\in \\Phi\\cup \\{1\\}$, $\\mathbb{T}$ acts on $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ by the character $\\bar{\\chi}\\sigma_{\\lambda}$ and on $Q_{\\sigma_{\\lambda}}$ by the character $\\sigma_{\\lambda}$. Since the characters $\\sigma_{\\lambda}$ are assumed to be distinct, it is easy to see that\n \\[P_{\\bar{\\chi}\\sigma_{\\lambda}}=\\{p\\in P| t\\cdot p=\\bar{\\chi}\\sigma_{\\lambda}(t)p\\text{ for }t\\in \\mathbb{T}\\}\\]\n \\[Q_{\\sigma_{\\lambda}}=\\{q\\in Q| t\\cdot q=\\sigma_{\\lambda}(t)q\\text{ for }t\\in \\mathbb{T}\\}.\\] The order of $\\mathbb{T}$ is coprime to $p$, hence Maschke's theorem asserts that any finite dimensional $\\mathbb{F}_p[\\op{G}']$-module $M$ decomposes into a direct sum $M=\\bigoplus_{\\tau} M_{\\tau}$, where $\\tau$ is a character of $\\mathbb{T}$ and $M_{\\tau}$ is the $\\tau$-eigenspace $M_{\\tau}:=\\{m\\in M| g\\cdot m=\\tau(g) m\\}$. Thus, we have the next Lemma, which follows from the discussion above.\n \\begin{Lemma}\\label{Pdecomposition} Let $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ and $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$ be Galois-stable subgroups.\n \\begin{enumerate}\n \\item As a $\\mathbb{T}$-module, $P$ decomposes into a direct sum of eigenspaces: \\[P=\\bigoplus_{\\lambda\\in \\Phi\\cup\\{1\\}} P_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\n \\item As a $\\mathbb{T}$-module, $Q$ decomposes into a direct sum of eigenspaces:\n \\[Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}.\\]\n \\end{enumerate}\n \\end{Lemma}\n \n \\begin{comment} We prove the statement for $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$. The proof is identical for $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. Let $\\Phi'$ denote $\\Phi\\cup \\{1\\}$. As $\\lambda$ ranges over $\\Phi'$, the characters $\\sigma_{\\lambda}$ are all assumed to be distinct. For $p\\in P$, represent \\[p=\\sum_{\\lambda\\in \\Phi'} p_{\\lambda}\\] where $p_{\\lambda}$ is the projection of $p$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi} \\sigma_{\\lambda}}$. For $\\lambda\\in \\Phi'$, let $P_{\\lambda}$ denote the projection of $P$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi}\\sigma_{\\lambda}}$. Let $\\Phi_0\\subset \\Phi'$ consist of $\\lambda$ such that the intersection $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ is not equal $P_{\\lambda}$. We show that $\\Phi_0$ is the empty set. Assume by way of contradiction that $\\Phi_0$ is not empty. Let $\\mathcal{P}$ consist of $p\\in P$ for which $p_{\\lambda}\\notin P$ for some $\\lambda\\in \\Phi_0$. Since $\\Phi_0$ is nonempty, so is $\\mathcal{P}$. For $p\\in \\mathcal{P}$, set $\\lambda_0(p)$ and $\\lambda_1(p)$ to be the minimal and maximal elements of $\\Omega_p:=\\{\\lambda\\in \\Phi'\\mid p_{\\lambda}\\notin P\\}$ respectively. By assumption, $p_{\\lambda}\\in P$ for $p\\notin \\Omega_p$. If $\\lambda_1(p)=\\lambda_0(p)$ then $p_{\\lambda_0}=p-\\sum_{\\lambda\\notin\\Omega_p}p_{\\lambda}$ is contained in $P$, which is clearly not the case as $\\lambda_0(p)$ is in $\\Omega_p$. Therefore, $\\lambda_0(p)$ is not equal to $\\lambda_1(p)$. Let $q\\in \\mathcal{P}$ be an element for which $\\lambda_0(p)\\leq \\lambda_0(q)$ for $p\\in \\mathcal{P}$. Set $\\lambda_i:=\\lambda_i(q)$ for $i=0,1$. It follows from conditions $\\eqref{thc3}$ and $\\eqref{thc4}$ of Theorem $\\ref{main}$, that there exists $g$ such that $\\bar{\\rho}(g)\\in \\mathcal{T}$ and $\\sigma_{\\lambda_0}(g)\\neq \\sigma_{\\lambda_1}(g)$. The element $q'$ defined by\n \\[q':=g\\cdot q-(\\bar{\\chi}\\sigma_{\\lambda_0})(g)q=\\sum_{\\lambda} \\left((\\bar{\\chi}\\sigma_{\\lambda})(g)-(\\bar{\\chi}\\sigma_{\\lambda_0})(g)\\right)q_{\\lambda}\\]\n is contained in $\\mathcal{P}$ and $\\lambda_1(q')=\\lambda_1$. Furthermore, it has been arranged that $q'_{\\lambda_0}=0$ and $q'_{\\lambda}$ is a scalar multiple of $q_{\\lambda}$ for every root $\\lambda$. Hence, we deduce that $\\lambda_0(q')>\\lambda_0$, which is a contradiction to the maximality of $\\lambda_0=\\lambda_0(q)$. Therefore $\\Phi_0$ is empty and the proof is complete.\\end{comment}\n Set the height of the formal symbol \"$1$\" to be equal to zero. Fix a total order on $\\Phi\\cup \\{1\\}$ such that $\\op{ht}(\\lambda)\\leq \\op{ht}(\\gamma)$ if $\\lambda\\leq \\gamma$. \n\\begin{Lemma}\\label{mainin}\n\\begin{enumerate} Let $P$ be a non-zero Galois stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Let $Q$ be a non-zero Galois stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}$. The following statements hold:\n \\item \\label{43c1} $P_{\\bar{\\chi}\\sigma_{2L_1}}$ is equal to $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$,\n \\item \\label{43c2} $Q_{\\sigma_{2L_1}}$ is equal to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nIt follows from Lemma $\\ref{Pdecomposition}$ that $P$ decomposes into $\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} P_{\\bar{\\chi}\\sigma_{\\lambda}}$. By condition $\\eqref{thc5}$ of Theorem $\\ref{main}$, the image of $\\sigma_{2L_1}$ spans $\\mathbb{F}_q$. Since $\\bar{\\chi}$, takes values in $\\mathbb{F}_p^{\\times}$, the same is true for the image of $\\bar{\\chi}\\sigma_{2L_1}$. Therefore, if $P_{\\bar{\\chi}\\sigma_{2L_1}}$ is not zero, then $P_{\\bar{\\chi}\\sigma_{2L_1}}=(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$. Suppose by way of contradiction that $P_{\\bar{\\chi}\\sigma_{2L_1}}=0$. Then we may choose $\\gamma\\in \\Phi\\cup \\{1\\}$ such that:\n\\begin{itemize}\n \\item $P_{\\bar{\\chi}\\sigma_{\\gamma}}\\neq 0$,\n \\item $P_{\\bar{\\chi}\\sigma_{\\lambda}}=0$ for all $\\lambda>\\gamma$.\n\\end{itemize} By assumption, $\\gamma$ is not the maximal root $2L_1$. There exists $\\gamma_1\\in \\Phi$ such that the difference $\\mu:=\\gamma_1-\\gamma$ is in $ \\Phi^{+}$. This can be shown by considering all possibilities for $\\gamma$:\n\\begin{enumerate}\n\\item $\\gamma=1$, then let $\\mu=\\gamma_1$ be any positive root,\n \\item $\\gamma=2L_i$ for $i> 1$, then $\\mu=L_{i-1}-L_i$ and $\\gamma_1=L_{i-1}+L_i$,\n \\item $\\gamma=-2L_i$ for $i>1$, then $\\mu=L_{i-1}+L_i$ and $\\gamma_1=L_{i-1}-L_i$,\n \\item $\\gamma=-2L_1$, then $\\mu=L_{1}+L_2$ and $\\gamma_1=-L_{1}+L_2$,\n \\item $\\gamma=L_i+L_j$ for $i2n$. Note that $\\mu$ is a positive root and hence, \n\\[g^{-1}\\cdot Y-Y=[X,Y]\\mod{(\\operatorname{Ad}^0\\bar{\\rho})_{\\op{ht}(\\mu)+k+1}}.\\]Note that since $-\\gamma_1+\\mu=-\\gamma$ is a root, \\[[(\\operatorname{Ad}^0\\bar{\\rho})_{\\mu},(\\operatorname{Ad}^0\\bar{\\rho})_{-\\gamma_1}]=(\\operatorname{Ad}^0\\bar{\\rho})_{-\\gamma}\\] (cf. \\cite[p. 39]{humphreys}). Letting $Y$ run through an appropriate basis of $\\operatorname{Ad}^0\\bar{\\rho}$, it follows from the above identity that $g\\cdot p-p$ can be expressed as a sum $a+b$ where $a\\neq 0$ is in $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}$ and $b\\in \\bigoplus _{\\lambda>\\gamma_1}P_{\\bar{\\chi}\\sigma_{\\lambda}}$. In particular, this shows that the projection of $g\\cdot p$ to $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}$ is non-zero.\n\\par Since $\\gamma_1=\\mu+\\gamma$ and $\\mu\\in \\Phi^+$, the height of $\\gamma_1$ is strictly larger than the height of $\\gamma$. As a result, $\\gamma_1>\\gamma$. Therefore, the subgroup $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}=0$. This contradiction shows that $\\gamma=2L_1$ and $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq 0$. This concludes part $\\eqref{43c1}$. The proof of part $\\eqref{43c2}$ is similar and is left to the reader.\n\\end{proof}\nFor $\\lambda\\in \\Phi\\cup \\{1\\}$, set \\[N_{\\lambda}=\\begin{cases}\n1\\text{ if }\\lambda\\in \\Delta,\\\\\n0\\text{ otherwise.}\n\\end{cases}\\]\n\\begin{Lemma}\\label{l2}\nLet $\\lambda\\in \\Phi\\cup \\{1\\}$ and $\\sigma_{\\lambda}$ the associated character. The following assertions are satisfied:\n\\begin{enumerate}\n\\item\\label{l2c1}\n$\\dim \\rm{Hom}( N, \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}=N_{\\lambda}$.\n\\item\\label{l2c2}\n$\\dim \\rm{Hom}(N', \\mathbb{F}_q(\\sigma_{\\lambda})^*)^{\\op{G}'\/N'}=0$.\n\\item\\label{l2c3} For $k\\neq 1$, \\[ H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0.\\]\nOn the other hand,\n\\[h^1 (\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2)= \\dim \\mathfrak{t}.\\]\n\\item\\label{l2c4} For all $k$,\n $h^1(\\op{G}', ((\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^*)=0$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nBy condition $\\ref{thc3}$ of Theorem $\\ref{main}$, $N$ may be identified with $U_1(\\mathbb{F}_q)$. The abelianization $N^{ab}$ is \\[U_1\/U_2(\\mathbb{F}_q)\\simeq \\bigoplus_{\\gamma\\in \\Delta} \\mathbb{F}_q(\\sigma_{\\gamma}).\\]\nBy condition $\\eqref{thc5}$ of Theorem $\\ref{main}$, any $\\op{G}\/N$ equivariant map $F:\\mathbb{F}_q(\\sigma_{\\gamma})\\rightarrow \\mathbb{F}_q(\\sigma_{\\lambda})$ is determined by the image of any nonzero element, hence \\[\\dim \\op{Hom}(\\mathbb{F}_q(\\sigma_{\\gamma}),\\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}\\leq 1.\\] We have that \\[F(\\sigma_{\\gamma}(g_1)\\sigma_{\\gamma}(g_2))=\\sigma_{\\lambda}(g_1)\\sigma_{\\lambda}(g_2)F(1)=F(\\sigma_{\\gamma}(g_1))F(\\sigma_{\\gamma}(g_2))F(1).\\] Since the image of $\\sigma_{\\lambda}$ spans $\\mathbb{F}_q$, it follows that $F$ is a scalar multiple of an element of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. By assumption, if $\\lambda\\neq \\gamma$, the characters $\\sigma_{\\lambda}$ and $\\sigma_{\\gamma}$ are not equal up to a twist of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. Therefore,\n\\[\\rm{Hom}( \\mathbb{F}_q(\\sigma_{\\gamma}), \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}=\\begin{cases} \\mathbb{F}_q \\mbox{ if $\\sigma_{\\lambda}=\\sigma_{\\gamma}$,}\\\\\n0 \\mbox{ otherwise,}\n\\end{cases}\\]and part $\\eqref{l2c1}$ follows this.\n\nObserve that $N'$ is isomorphic to $N$ and $\\op{G}'\/N'$ is the Galois group $\\op{Gal}(\\mathbb{Q}(\\{\\varphi_i\\}, \\bar{\\kappa},\\bar{\\chi})\/\\mathbb{Q})$. By condition $\\eqref{thc4}$, the characters $\\sigma_{\\gamma}$ and $\\sigma_{\\lambda}^*=\\bar{\\chi}\\sigma_{-\\lambda}$ are not equivalent up to a twist of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. Part $\\eqref{l2c2}$ follows via the same reasoning as part $\\eqref{l2c1}$.\n\\par The order of $\\op{G}\/N$ is coprime to $p$. By inflation-restriction and part $(\\ref{l2c1})$\n\\begin{equation*}\n\\begin{split}\n& \\dim H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})\\\\\n= & \\dim \\rm{Hom}(N, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^{\\op{G}\/N}\\\\\n= & \\sum_{\\lambda\\in\\Phi, ht(\\lambda)=k} \\dim \\rm{Hom}(N, \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}\\\\\n= & \\sum_{\\lambda\\in \\Phi, ht(\\lambda)=k} N_{\\lambda}.\\\\\n\\end{split}\n\\end{equation*}\n\n\nIt follows that if $k\\neq 1$, \n\\[ H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0\\]and that\n\\[h^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_{2})=\\# \\Delta =n=\\dim \\mathfrak{t}.\\] This concludes part $\\eqref{l2c3}$.\n\\par The order of $\\op{G}'\/N'$ is coprime to $p$. Therefore, by inflation-restriction,\n\\begin{equation*}\n\\begin{split}\n& \\dim H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})\\\\\n= & \\dim \\rm{Hom}(N', (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^{\\op{G}'\/N'}\\\\\n= & \\sum_{\\lambda\\in\\Phi, ht(\\lambda)=k} \\dim \\rm{Hom}(N', \\mathbb{F}_q(\\sigma_{\\lambda})^*)^{\\op{G}'\/N'}\\\\\n=& 0.\n\\end{split}\n\\end{equation*} This concludes the proof of part $\\eqref{l2c4}$.\n\\end{proof}\n\\begin{Def}\\label{perpdef}\nLet $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\subset \\operatorname{Ad}^0\\bar{\\rho}^*$ be the subspace of $\\operatorname{Ad}^0\\bar{\\rho}^*$ consisting of $f\\in \\operatorname{Ad}^0\\bar{\\rho}^*$ for which $f_{\\restriction (\\operatorname{Ad}^0\\bar{\\rho})_k}=0$.\n\\end{Def}\n\\begin{Remark}\nFor $k>-2n+1$, the submodule $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\neq 0$ and by Lemma $\\ref{mainin}$, \\[(\\operatorname{Ad}^0\\bar{\\rho})_{k,\\bar{\\chi}\\sigma_{2L_1}}^{\\perp}\\simeq(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi}\\sigma_{2L_1}}^*.\\] \n\\end{Remark}\n\\begin{Lemma}\\label{l3}\nLet $k$ be an integer,\n\\begin{enumerate}\n\\item\\label{l3c1}\n$H^1(\\op{G},\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$ and $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$,\n\\item\\label{l3c2}\n$H^1(\\op{G}',(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=0$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nWe begin with the proof of part $\\eqref{l3c1}$. Consider the case when $k\\leq 1$. By part $\\eqref{l2c3}$ of Lemma $\\ref{l2}$, for $i\\leq 1$, \n\\begin{equation*}\nH^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})=0.\n\\end{equation*}\nand hence there is an injection\n\\begin{equation*}H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_i)\\hookrightarrow H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i-1}).\\end{equation*}\nWe deduce that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\n\\par Next consider the case $k>1$. Associated to \n\\begin{equation*}\n0\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1\\rightarrow 0 \\end{equation*}\nis the long exact sequence in cohomology. It follows from $\\ref{mainin}$ that any non-zero submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ has a non-trivial $\\sigma_{2L_1}$ eigenspace for the $\\mathbb{T}$-action. As a consequence, $H^0(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})=0$. Since Lemma $\\ref{l2}$ asserts that \\[H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0,\\] we have that $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})$ surjects onto $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})$ for $k>1$. As a result, $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho})$ surjects onto $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})$, and therefore, \\[H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})=0.\\]\n\\par Since it has been shown that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)=0$, we have a short exact sequence\n\\[0\\rightarrow H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\rightarrow H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow 0.\\]It suffices to show that\n\\[\\dim H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\geq \\dim H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k).\\]\nCondition $\\eqref{thc4}$ of Theorem $\\ref{main}$ stipulates that for $\\lambda \\in \\Phi$, $\\sigma_{\\lambda}$ is not equal to $\\sigma_1=1$. Therefore for $i\\leq 0$,\n\\begin{equation*}\nH^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})=\\bigoplus_{ \\text{ht} \\gamma=i-1} H^0(\\op{G}, \\mathbb{F}_q(\\sigma_{\\gamma}))=0.\n\\end{equation*}\nFor $i\\leq 0$, we deduce that\n\\begin{equation*}\nH^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_i\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\xrightarrow{\\sim} H^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nComposing these isomorphisms we have that\n\\begin{equation*}\n(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=H^0(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\xrightarrow{\\sim} H^0(\\op{G},\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nWe have deduced that\n\\begin{equation*}\nh^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=\\dim \\mathfrak{t}.\n\\end{equation*}\nBy Lemma $\\ref{l2}$ part $\\eqref{l2c3}$,\n\\begin{equation*}\nh^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2)=\\dim \\mathfrak{t}=h^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nBy Lemma $\\ref{l2}$, for $i\\geq 2$, we have that \\begin{equation*}H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_i\/(\\operatorname{Ad}^0\\bar{\\rho})_{i+1})=0.\\end{equation*} and it follows that $H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_2\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\nHence it follows that\n\\begin{equation}\\label{deltaiso}h^1(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\leq h^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2).\\end{equation}We conclude that\n\\[h^1(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\leq h^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/ (\\operatorname{Ad}^0\\bar{\\rho})_1).\\]Therefore we conclude that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\n\\par Since $[L:K]$ is coprime to $p$, from a direct application of inflation-restriction it follows that $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\nWe have proved part $(\\ref{l3c1})$.\n\\par Consider the short exact sequence \\[0\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}^{\\perp}\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_i^{\\perp} \\rightarrow ((\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})^*\\rightarrow 0\\] and the associated sequence in cohomology. By Lemma $\\ref{l2}$, \n\\begin{equation*}H^1(\\op{G}', ((\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})^*)=0\n\\end{equation*}\nfrom which we deduce that \n\\begin{equation*}\nH^1(\\op{G}',(\\operatorname{Ad}^0\\bar{\\rho})_{i-1}^{\\perp})\\rightarrow H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_i^{\\perp})\n\\end{equation*}\nis a surjection for all $i$. As \\[(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+1}^{\\perp}\\simeq (\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+1})^*=0\\] we deduce that $H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=0$.\n\\end{proof}\n\\par For $\\psi$ in $H^1(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, the restriction $\\psi_{\\restriction \\op{G}_K}:\\op{G}_K\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ is a homomorphism since the action of $\\op{G}_K$ on $\\operatorname{Ad}^0\\bar{\\rho}^*$ is trivial. Let $K_{\\psi}\\supseteq K$ be the extension \\textit{cut out} by $\\psi$, i.e. $K_{\\psi}$ is the smallest extension of $K$ which is fixed by the kernel of $\\psi_{\\restriction \\op{G}_K}$. Identify $\\op{Gal}(K_{\\psi}\/K)$ with $\\psi(\\op{G}_K)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ and let $J_{\\psi}\\subseteq K_{\\psi}$ be the subfield for which $\\op{Gal}(K_{\\psi}\/J_{\\psi})\\simeq \\psi(\\op{G}_K)_{\\bar{\\chi}\\sigma_{2L_1}}$. Likewise for $f$ in $H^1(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})$ denote by $L_f$, the extension of $L$ cut out by $f$. Set $K_f$ to be the composite of $L_f$ with $K$. Since $p\\nmid [K:L]$, we have that $\\op{Gal}(K_f\/K)\\simeq \\op{Gal}(L_f\/L)$.\n\\begin{Lemma}\\label{l4}Let $\\mathcal{J}\\supseteq S$ be a finite set of primes. Let $f\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$ and $\\psi\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$ be a non-zero cohomology classes. Then the following assertions are satisfied:\n\\begin{enumerate}\n\\item\\label{l4p1} $L_{f}\\supsetneq L$ (equivalently, $K_f\\supsetneq K$),\n\\item\\label{l4p2}\n$K_{\\psi}\\supsetneq J_{\\psi}$, in particular, $K_{\\psi}\\supsetneq K$.\n\\end{enumerate}\n\\begin{proof}\n\\par For part $\\eqref{l4p2}$, recall that Lemma $\\ref{l3}$ asserts that\n$H^1(\\op{G}', \\operatorname{Ad}^0\\bar{\\rho}^*)=0$. Therefore, the restriction $\\psi_{\\restriction \\op{G}_K}$ is not zero. This shows that $K_{\\psi}\\supsetneq K$. That $K_{\\psi}$ strictly contains $J_{\\psi}$ is a direct consequence of Lemma $\\ref{mainin}$. Part $\\eqref{l4p1}$ also follows from Lemma $\\ref{l3}$.\n\\end{proof}\n\\end{Lemma}\n\n\\begin{Lemma}Let $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ (resp. $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$) be a nonzero Galois-stable subgroup and $\\iota_P:P\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ (resp. $\\iota_Q:Q\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}$) denote the inclusion. Then, \\label{y1}\n\\begin{enumerate}\n \\item\\label{48c1} \n$\\rm{Hom}_{\\mathbb{F}_p}(P,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=\\mathbb{F}_q\\cdot \\iota_P$,\n\\item\\label{48c2} \n$\\rm{Hom}_{\\mathbb{F}_p}(Q,\\operatorname{Ad}^0\\bar{\\rho}^*)^{G}=\\mathbb{F}_q\\cdot \\iota_Q$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nWe prove part $\\eqref{48c1}$, the proof of $\\eqref{48c2}$ is similar. Let $\\Phi'=\\Phi\\cup \\{1\\}$ and $m:=\\# \\Phi'$. Enumerate $\\Phi'=\\{\\gamma_1,\\dots, \\gamma_m\\}$, so that $\\gamma_i>\\gamma_j$ if $ii$ be such that $\\varphi(p)\\in W_j$ and $\\varphi(p)\\notin W_{j-1}$. It follows from conditions $\\eqref{thc3}$ and $\\eqref{thc4}$ of Theorem $\\ref{main}$, that there exists $g\\in \\mathbb{T}$ such that $\\sigma_{\\gamma_i}(g)\\neq \\sigma_{\\gamma_j}(g)$. Express $\\varphi(p)=x_j+x_{j-1}$, where $x_{j}\\in (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi} \\sigma_{\\gamma_j}}$ and $x_{j-1}\\in W_{j-1}$. We have that \n\\[g\\varphi(p)=\\bar{\\chi}(g)\\sigma_{\\gamma_j}(g)x_{j}+g\\cdot x_{j-1}=\\varphi(gp)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)(x_j+x_{j-1}).\\]\nWe have that\n\\[\\bar{\\chi}(g)(\\sigma_{\\gamma_j}(g)-\\sigma_{\\gamma_i}(g))x_{j}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)x_{j-1}-g\\cdot x_{j-1}\\]is contained in $W_{j-1}$. This is contradiction, we deduce that $\\varphi(P_i)\\subseteq W_i$. We show that $\\varphi(p)=\\beta p$ for $p\\in P_{\\bar{\\chi}\\sigma_{\\gamma_i}}$ and this shall complete the inductive step. Write $\\varphi(p)=z_i+z_{i-1}$ where $z_i\\in (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\gamma_i}}$ and $z_{i-1}\\in W_{i-1}$. Let $g\\in \\mathbb{T}$, we have that \\[\\varphi(g p)=\\varphi(\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) p)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)\\varphi( p)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)z_i+\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) z_{i-1}.\\] We have that\n\\[g\\varphi(p)=g\\cdot z_i+g \\cdot z_{i-1}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)z_i+g\\cdot z_{i-1}.\\]\nTherefore, we deduce that \n\\[g\\cdot z_{i-1}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) z_{i-1}.\\] We show that $z_{i-1}=0$. If $z_{i-1}\\neq 0$, there exists $k1$, we have that ${P'}_{\\bar{\\chi}\\sigma_{2L_1}}=0$ and therefore Lemma $\\ref{mainin}$, asserts that $P'=0$. We conclude that $\\varphi(p)=z_i=\\beta p$. This concludes the induction step and the proof of the result.\n\\end{comment}\n\\end{proof}\n\\begin{Cor}\\label{Coradd}\nThe following statements hold:\n\\begin{enumerate}\n \\item\\label{coradd1}\n let $P_1$ and $P_2$ be Galois-stable subgroups of $\\operatorname{Ad}^0\\bar{\\rho}^*$ such that there is an isomorphism $\\phi:P_1\\xrightarrow{\\sim} P_2$ of Galois modules. Then $P_1=P_2$ and $\\phi$ is multiplication by a scalar.\n \\item \\label{coradd2}\n Let $Q_1$ and $Q_2$ be Galois-stable subgroups of $\\operatorname{Ad}^0\\bar{\\rho}$ such that there is an isomorphism $\\phi:Q_1\\xrightarrow{\\sim} Q_2$ of Galois modules. Then $Q_1=Q_2$ and $\\phi$ is multiplication by a scalar.\n\\end{enumerate}\n\\end{Cor}\n\\begin{proof}\nWe prove part $\\eqref{coradd1}$, part $\\eqref{coradd2}$ is identical. Let $\\iota_{P_i}:P_i\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ be the inclusion. By Proposition $\\ref{y1}$, the two inclusions $\\iota_{P_1}$ and $\\iota_{P_2}\\circ \\phi$ are the same upto a scalar. The assertion follows.\n\\end{proof}\nLet $Q$ be a $\\op{G}$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$, by Lemma $\\ref{Pdecomposition}$, the projection of $Q$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ equals $Q_{\\sigma_{-2L_1}}$. For convenience of notation, let $Q_{-2L_1}$ denote $Q_{\\sigma_{-2L_1}}$.\n\\begin{Lemma}\\label{fullrankLemma}\nLet $Q$ be a Galois-stable submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ for which $ Q_{-2L_1}\\neq 0$, then $Q=\\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{Lemma}\n\\begin{proof}\nLet $P:=\\{\\gamma\\in \\operatorname{Ad}^0\\bar{\\rho}^*\\mid \\gamma(x)=0 \\text{ for }x\\in Q\\}$. The assumption on $Q$ implies that $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$. Since the image of $\\bar{\\chi}\\sigma_{2L_1}$ spans $\\mathbb{F}_q$, $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$ implies that $P_{\\bar{\\chi}\\sigma_{2L_1}}=0$ By Lemma $\\ref{mainin}$, $P=0$, and therefore, $Q=\\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{proof}\n\n\\begin{Lemma}\\label{22Dec5} The following statements hold:\n\\begin{enumerate}\n \\item\\label{411c1} the fields $K=\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ and $\\mathbb{Q}(\\mu_{p^2})$ are linearly disjoint over $\\mathbb{Q}(\\mu_p)$.\n \\item\\label{411c2} Let $\\mathcal{J}\\supseteq S$ be a finite set of prime numbers, $\\psi_1,\\dots, \\psi_t\\in H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}},\\operatorname{Ad}^0\\bar{\\rho}^*)$ and set $K_j:=K_{\\psi_j}$ for $j=1,\\dots, t$. Then the composite $K_1\\cdots K_t$ and $\\mathbb{Q}(\\mu_{p^2})$ are linearly disjoint over $\\mathbb{Q}(\\mu_p)$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nSuppose by way of contradiction that $\\mathbb{Q}(\\mu_{p^2})\\subseteq K$. Set $V:=\\op{Gal}(K\/F(\\mu_{p^2}))$ and $\\mathcal{A}:=\\op{G}'\/N'=\\op{Gal}(F(\\mu_p)\/\\mathbb{Q})$. For $n\\in N'^{ab}$ and $g\\in \\mathcal{A}$, let $\\tilde{n}$ and $\\tilde{g}$ be lifts of $n$ and $g$ to $N'$ and $\\op{G}'$ respectively. The action of $\\mathcal{A}$ on $N'^{ab}$ is induced by conjugation, defined by $g\\cdot n:= \\tilde{g}\\tilde{n}\\tilde{g}^{-1}\\mod{[N',N']}$. The groups $N$ and $N'$ are isomorphic and the image of $\\bar{\\rho}$ is assumed to contain $U_1(\\mathbb{F}_q)$ (condition $\\ref{thc3}$ of Theorem $\\ref{main}$). The quotient $N'\/V=\\op{Gal}(F(\\mu_{p^2})\/F(\\mu_p))\\simeq \\mathbb{F}_p$. Let $\\pi: \\op{N}'^{ab}\\rightarrow \\mathbb{F}_p$ denote the map induced by the mod-$V$ quotient. Being the composite of Galois extensions, $F(\\mu_{p^2})$ is Galois over $\\mathbb{Q}$. As a result, $\\pi$ is $\\mathcal{A}$-equivariant. Furthermore, since $F(\\mu_{p^2})$ is an abelian extension of $\\mathbb{Q}$, the $\\mathcal{A}$-action on $N'\/V$ is trivial. On the other hand, as an $\\mathcal{A}$-module, $N'^{ab}\\simeq \\bigoplus_{\\lambda\\in \\Delta} \\mathbb{F}_q(\\sigma_{\\lambda})$. It follows from condition $\\ref{thc4}$ of Theorem $\\ref{main}$ that $\\sigma_{\\lambda}\\neq 1$ for $\\lambda\\in \\Phi$. As a result, \n\\[\\op{Hom}(N'^{ab}, \\mathbb{F}_p)^{\\op{G}'}\\simeq \\bigoplus_{\\lambda\\in \\Delta}\\op{Hom}(\\mathbb{F}_q(\\sigma_{\\lambda}), \\mathbb{F}_p)^{\\op{G}'}=0.\\] This is a contradiction which concludes the proof of the first part.\n\\par \nSet $\\mathcal{K}_j$ to be $K_1\\dots K_j$ and $\\mathcal{K}_0:=K$. Setting $E:=\\mathbb{Q}(\\mu_{p^2})$, it suffices to show that $\\mathcal{K}_j\\cap E=\\mathcal{K}_{j-1}\\cap E$. We begin with the case $j=1$. For $\\psi\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, regard $\\operatorname{Gal}(K_{\\psi}\/K)$ as an $\\mathbb{F}_q[\\op{G}']$-module, where the Galois action is induced via conjugation. The $\\op{G}'$-module $P_1:=\\operatorname{Gal}(K_{1}\/K)$ is identified with $\\psi_1(\\op{G}_K)$. Let $Q_1\\subseteq P_1$ be the $\\op{G}'$-stable subgroup defined by $Q_1:=\\op{Gal}(K_1\/(K_1\\cap E)\\cdot K)$. The action of $\\op{G}'$ on $P_1\/Q_1=\\op{Gal}((K_1\\cap E)\\cdot K\/K)$ is trivial. By Lemma $\\ref{Pdecomposition}$, the quotient $P_1\/Q_1$ decomposes into subgroups \n\\[P_1\/Q_1=\\bigoplus_{\\lambda\\in \\Phi\\cup\\{1\\}} (P_1)_{\\bar{\\chi}\\sigma_{\\lambda}}\/(Q_1)_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\nThe characters $\\sigma_{\\lambda}\\neq \\bar{\\chi}^{-1}$ and hence $P_1=Q_1$. We have thus shown that $K_1\\cap E= K\\cap E$.\n\\par Let $P_j$ be defined by $P_j:=\\operatorname{Gal}(\\mathcal{K}_j\/\\mathcal{K}_{j-1})$. The $\\op{G}'$-module $P_j$ is isomorphic to \\[ \\operatorname{Gal}(K_j\/K_j\\cap \\mathcal{K}_{j-1})\\subseteq \\psi_j(\\op{G}_K)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*.\\] Let $Q_j$ be the $\\op{G}'$-stable subgroup $\\op{Gal}(\\mathcal{K}_j\/(\\mathcal{K}_j\\cap E)\\cdot \\mathcal{K}_{j-1})$ and note that the $\\op{G}'$ action on $P_j\/Q_j$ is trivial. Invoking the same argument as in the case when $j=1$, we have that $P_j=Q_j$ and hence $\\mathcal{K}_j\\cap E=\\mathcal{K}_{j-1}\\cap E$. This completes the proof. \n\\end{proof}\n\n\n\n\\begin{Def}\n\n\\begin{enumerate}\n \\item Let $M_1$ and $M_2$ be $\\mathbb{F}_p[\\op{G}']$-modules. We say that $M_1$ is unrelated to $M_2$ if for every $\\mathbb{F}_p[\\op{G}']$-submodule $N$ of $M_1$, \n\\[\\op{Hom}(N,M_2)^{\\op{G}'}=0.\\]\n\n\\item Let $E$ be a finite extension of $K$ such that $E$ is Galois over $\\mathbb{Q}$ and $\\op{Gal}(E\/K)$ is an $\\mathbb{F}_p$-vector space. Let $M$ be an $\\mathbb{F}_p[\\op{G}']$-module. We say that $E$ is unrelated to $M$ if $\\op{Gal}(E\/K)$ is $\\op{G}'$-unrelated to $M$. Here, the $\\op{G}'$-action on $\\op{Gal}(E\/K)$ is induced via conjugation (let $x\\in \\op{Gal}(E\/K)$ and $g\\in \\op{G}'$, pick a lift $\\tilde{g}$ of $g$, set $g\\cdot x:=\\tilde{g}x\\tilde{g}^{-1}$).\n\\end{enumerate}\n\n\\end{Def}\n\\begin{Prop}\\label{414}\nLet $\\mathcal{J}\\supseteq S$ be a finite set of prime numbers and \\[\\theta_0,\\dots, \\theta_t\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\] be linearly independent over $\\mathbb{F}_q$. Set $K_i:=K_{\\theta_i}$ and let $\\mathbb{L}_1,\\dots, \\mathbb{L}_k$ be a (possibly empty) set of Galois extensions of $\\mathbb{Q}$. Assume that $\\mathbb{L}_i$ contains $K$ and $\\op{Gal}(\\mathbb{L}_i\/K)$ is an $\\mathbb{F}_p$-vector space for $i=1,\\dots, k$. Suppose that $\\mathbb{L}_i$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$ for $i=1,\\dots, k$. Denote by $\\mathcal{L}$ the composite $\\mathbb{L}_1\\cdots \\mathbb{L}_k$. If the set $\\{\\mathbb{L}_1,\\dots, \\mathbb{L}_k\\}$ is empty, set $\\mathcal{L}=K$. The field $K_0$ is not contained in the composite of the fields $K_1\\cdots K_t \\cdot \\mathcal{L}$.\n\\end{Prop}\n\\begin{proof}\nLet $\\mathcal{K}$ denote the composite of the fields $K_1,\\dots, K_t$. If $K_0$ is contained in $\\mathcal{K}\\cdot \\mathcal{L}$, then $\\theta_0,\\dots, \\theta_t\\in H^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)$ and hence\n\\[h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\geq t+1.\\] Hence it suffices to show that \n\\[h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq t.\\]First we show that \n\\[h^1(\\op{Gal}( \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)=0.\\] Denote by $\\mathcal{L}_i$ the composite of the fields $\\mathbb{L}_1\\cdots \\mathbb{L}_i$ and set $\\mathcal{L}_0:=K$. Note that $\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})$ is isomorphic to $\\op{Gal}(\\mathbb{L}_i\/\\mathbb{L}_i\\cap \\mathcal{L}_{i-1})$, which is an $\\mathbb{F}_p[\\op{G}']$-submodule of $\\op{Gal}(\\mathbb{L}_i\/K)$. Since $\\mathbb{L}_i$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n\\[\\rm{Hom}(\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1}),\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=0.\\]Hence the inflation map\n\\[H^1(\\op{Gal}(\\mathcal{L}_{i-1}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\xrightarrow{\\op{inf}} H^1(\\op{Gal}(\\mathcal{L}_{i}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\]is an isomorphism. We deduce that $H^1(\\op{Gal}(\\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)$ is isomorphic to $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}^*)$ and hence, is zero.\n\\par Let $\\mathcal{K}_i$ denote the composite $K_1\\cdots K_i$ and $\\mathcal{K}_0$ denote $K$. Note that $\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathcal{K}_{i-1}\\cdot\\mathcal{L})$ is an $\\mathbb{F}_p[\\op{G}']$-submodule of $\\op{Gal}(K_i\/K)$, and hence, of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Lemma $\\ref{y1}$ asserts that\n\\[\\dim \\rm{Hom}(\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathcal{K}_{i-1}\\cdot\\mathcal{L}),\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1.\\]Therefore, by inflation-restriction,\n\\[h^1(\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq h^1(\\op{Gal}(\\mathcal{K}_{i-1}\\cdot\\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*) +1.\\]Consequently, we deduce that\n$h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq t$ and the proof is complete.\n\\end{proof}\n\\begin{Lemma}\\label{415} Let $\\mathcal{J}\\supseteq S$ be a finite set of primes.\n \\begin{enumerate}\n \n \\item\\label{415c1} Let $M$ be a nontrivial quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ and $\\eta\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}},M)$ be non-zero. Let $K_{\\eta}$ be the field extension of $K$ cut out by $\\eta$. The field $K_{\\eta}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item \\label{415c2} The field $K(\\mu_{p^2})$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item\\label{415c3} Let $f\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$, then the extension $K_f$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item \\label{415c4} Suppose that we are given a lift $\\zeta_2:\\op{G}_{\\mathbb{Q}, \\mathcal{J}}\\rightarrow \\op{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$ of $\\bar{\\rho}$ with similitude character $\\kappa\\mod{p^2}$. The field extension $K(\\zeta_2)$ cut out by the kernel of $\\zeta_2$, is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n\\par For part $\\eqref{415c1}$, it suffices to show that $M$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$. Since $M$ is a non-trivial quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$, it follows from Lemma $\\ref{mainin}$ that the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $M$ is zero. Let $N\\subseteq M$ be a $\\op{G}'$-submodule and $f:N\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ be a homomorphism. The $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $f(N)$ is zero, hence by Lemma $\\ref{mainin}$, the map $f=0$.\n\\par Since $\\mathbb{Q}(\\mu_{p^2})$ is an abelian extension of $\\mathbb{Q}$, the $G'$-action on $\\op{Gal}(K(\\mu_{p^2})\/K)$ is trivial. On the other hand, $\\operatorname{Ad}^0\\bar{\\rho}^*$ has no trivial $\\mathbb{T}$-eigenspace. It follows that $K(\\mu_{p^2})$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$ and part $\\eqref{415c2}$ follows.\n\\par \nLet $Q$ be a $\\op{G}'$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$, Lemma $\\ref{Pdecomposition}$ asserts that $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$. On the other hand, $\\operatorname{Ad}^0\\bar{\\rho}^*=\\bigoplus_{\\gamma\\in \\Phi\\cup \\{1\\}} (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$. It follows from condition $\\eqref{thc4}$ of Theorem $\\ref{main}$ that \n\\[\\op{Hom}(Q_{\\sigma_{\\lambda}}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\gamma}})^{\\mathbb{T}}=0.\\] As a result, $\\op{Hom}(Q,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=0$ and hence part $\\eqref{415c3}$ follows.\n\\par For part \\eqref{415c4}, identify $\\operatorname{Ad}^0\\bar{\\rho}$ with the kernel of the mod-$p$ reduction map \\[\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\op{Sp}_{2n}(\\mathbb{F}_q),\\]by identifying $X\\in \\operatorname{Ad}^0\\bar{\\rho}$ with $\\op{Id}+pX$. Recall that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer which is divisible by $p(p-1)$. Setting $\\kappa_2:=\\kappa\\mod{p^2}$, we see that the restriction $\\kappa_{2\\restriction \\op{G}_K}$ is trivial. Therefore, $\\op{Gal}(K(\\zeta_2)\/K)$ may be identified with a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Here, $g\\in \\op{Gal}(K(\\zeta_2)\/K)$ is identified with \n\\[\\zeta_2(g)\\in \\op{ker}\\left(\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\op{Sp}_{2n}(\\mathbb{F}_q)\\right)\\simeq \\operatorname{Ad}^0\\bar{\\rho}.\\] The same reasoning as in the previous case shows that $K(\\zeta_2)$ is indeed unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\end{proof}\n\\begin{Lemma}\\label{lemma416}\nLet $L_1,\\dots, L_k$ and $K_1,\\dots , K_l$ be Galois extensions of $\\mathbb{Q}$ which contain $K$. Assume that:\n\\begin{itemize}\n \\item $\\op{Gal}(L_i\/K)$ and $\\op{Gal}(K_i\/K)$ are finite dimensional $\\mathbb{F}_p$-vector spaces.\n \\item As a $\\op{G}'$-module, $\\op{Gal}(L_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}$ for $i=1,\\dots, k$.\n \\item As a $\\op{G}'$-module, $\\op{Gal}(K_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ for $i=1,\\dots, l$.\n\\end{itemize} Then the composite $L_1\\cdots L_k$ is linearly disjoint from $K_1,\\dots, K_l$.\n\\end{Lemma}\n\\begin{proof}\nThe order of $\\mathbb{T}$ is coprime to $p$, hence Maschke's theorem asserts that any finite dimensional $\\mathbb{F}_p[\\op{G}']$-module $M$ decomposes into a direct sum \n\\[M=\\oplus_{\\tau} M_{\\tau}.\\]Here, $\\tau$ is a character of $\\mathbb{T}$ and $M_{\\tau}$ is the $\\tau$-eigenspace \n\\[M_{\\tau}:=\\{m\\in M| g\\cdot m=\\tau(g) m\\}.\\]The action of $\\op{G}'$ on $\\op{Gal}(L_i\/K)$ and $\\op{Gal}(K_i\/K)$ is induced by conjugation. By assumption, $\\op{Gal}(L_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}$, i.e. there exist $\\op{G}'$-submodules $Q_1\\subseteq Q_2$ of $\\operatorname{Ad}^0\\bar{\\rho}$ such that $\\op{Gal}(L_i\/K)\\simeq Q_2\/Q_1$. By Lemma $\\ref{Pdecomposition}$, the module $Q_i$ decomposes into $\\mathbb{T}$-eigenspaces\n\\[Q_i=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(Q_i)_{\\sigma_{\\lambda}}\\]for $i=1,2$. Therefore, the quotient $\\op{Gal}(L_i\/K)$ decomposes into \\[\\op{Gal}(L_i\/K)=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(\\op{Gal}(L_i\/K))_{\\sigma_{\\lambda}}\\] where $(\\op{Gal}(L_i\/K))_{\\sigma_{\\lambda}}:=(Q_2)_{\\sigma_{\\lambda}}\/(Q_1)_{\\sigma_{\\lambda}}$ is the $\\sigma_{\\lambda}$-eigenspace for the action of $\\mathbb{T}$ on $\\op{Gal}(L_i\/K)$. Likewise, $\\op{Gal}(K_i\/K)$ decomposes into \\[\\op{Gal}(K_i\/K)=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(\\op{Gal}(K_i\/K))_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\n\\par Let $\\mathcal{L}$ be the composite $L_1\\cdots L_k$ and $\\mathcal{K}$ be the composite $K_1\\cdots K_l$. Letting $\\mathcal{L}_i$ be the composite $L_1\\cdots L_i$, filter $\\mathcal{L}$ by\n \\[\\mathcal{L}\\supseteq \\mathcal{L}_{k-1}\\cdots \\supseteq \\mathcal{L}_1\\supseteq K.\\] The Galois group \\[\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})\\simeq \\op{Gal}(L_i\/L_i\\cap \\mathcal{L}_{i-1})\\] is a $\\op{G}'$-submodule of $\\op{Gal}(L_i\/K)$. Hence the characters for the action of $\\mathbb{T}$ on $\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})$ are each of the form $\\sigma_{\\lambda}$. Similar reasoning shows that the characters for the action of $\\mathbb{T}$ on $\\op{Gal}(\\mathcal{K}\/K)$ are each of the form $\\bar{\\chi}\\sigma_{\\lambda}$. Set $E=\\mathcal{K}\\cap \\mathcal{L}$ and $M=\\op{Gal}(E\/K)$. Being a quotient of $\\op{Gal}(\\mathcal{L}\/K)$, $M$ decomposes into eigenspaces for the action of the torus \n\\[M=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} M_{\\sigma_{\\lambda}}.\\]Since $M$ is a quotient of $\\op{Gal}(\\mathcal{K}\/K)$, \\[M=\\bigoplus_{\\gamma\\in \\Phi\\cup \\{1\\}} M_{\\bar{\\chi}\\sigma_{\\gamma}}.\\] It is assumed that the image of $\\sigma_{\\lambda}$ spans $\\mathbb{F}_q$ and that $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$ twist of $\\bar{\\chi}\\sigma_{\\gamma}$. Hence, it follows that\n\\[\\op{Hom}(\\mathbb{F}_q(\\sigma_{\\lambda}),\\mathbb{F}_q(\\bar{\\chi}\\sigma_{\\gamma}))^{\\mathbb{T}}=0.\\]Therefore, $\\rm{Hom}(M,M)^{\\op{G}'}=0$ and in particular, the identity map is zero. This implies that $\\mathcal{K}\\cap \\mathcal{L}=K$.\n\\end{proof}\n\\section{Deformation conditions at Auxiliary Primes}\n\nWe introduce the auxiliary primes $v$ and the liftable deformation problem $\\mathcal{C}_v$ at $v$.\n\\begin{Def} \nA prime number $v$ is a trivial prime if the following splitting conditions are satisfied:\n\\begin{itemize}\n\\item $\\operatorname{G}_v\\subseteq \\ker\\bar{\\rho}$,\n\n\\item $v\\equiv 1 \\mod{p}$ and $v \\not\\equiv 1 \\mod{p^2}$.\\end{itemize} \n\\end{Def}\n In other words, a prime number $v$ is trivial if it splits in $\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ and does not split in $\\mathbb{Q}(\\mu_{p^2})$. By Lemma $\\ref{22Dec5}$, $\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ does not contain $\\mathbb{Q}(\\mu_{p^2})$. This is a Chebotarev condition, i.e. defined by a finite union of sets that are defined by applying the Chebotarev density theorem. Therefore, the set of trivial primes has positive Dirichlet density, in particular, it is infinite.\\par Let $v$ be a trivial prime. The deformations of the trivial representation $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ are tamely ramified. The Galois group of the maximal pro-p extension of $\\mathbb{Q}_{v}$ is generated by a Frobenius $\\sigma_v$ and a generator of tame pro-$p$ inertia $\\tau_v$. These satisfy the relation \n$\\sigma_v\\tau_v\\sigma_v^{-1}=\\tau_l^{v}$. We define the deformation functor $\\mathcal{C}_v$. The functor $\\mathcal{C}_v$ will be liftable, however, it will not be a deformation condition. Let $\\alpha$ be a root which shall be specified later. The root-subgroup $\\text{U}_{\\alpha}\\subset \\operatorname{GSp}_{2n}$ is the subgroup generated by the image of the root-subspace $(\\op{sp}_{2n})_{\\alpha}$ under the exponential map. We let $\\text{Z}(\\text{U}_{\\alpha})$ be the subgroup of $\\operatorname{GSp}_{2n}$ consisting of elements which commute with $\\text{U}_{\\alpha}$.\n\n\n\\begin{Def}\\label{ramtriv}\\cite[Definition 3.1]{FKP1}\n Let $\\mathcal{D}_v^{\\alpha}$ consist of the deformation classes of lifts such that some representative $\\varrho$ satisfies:\n \\begin{enumerate}\n \\item $\\varrho(\\sigma_v)\\in \\mathcal{T}\\cdot \\text{Z}(\\text{U}_{\\alpha})$ and $\\varrho(\\tau_v)\\in \\text{U}_{\\alpha}$,\n \\item under the composite \n \\[\\mathcal{T}\\cdot \\text{Z}(\\text{U}_{\\alpha})\\rightarrow \\mathcal{T}\/(\\mathcal{T}\\cap \\text{Z}(\\text{U}_{\\alpha}))\\xrightarrow{\\alpha} \\text{GL}_1\\] $\\varrho(\\sigma_v)$ maps to $v$.\n \\end{enumerate}\n \\begin{Remark}\n When $n=1$ and $\\alpha$ is the positive root of $\\operatorname{sl}_2$, the deformation functor $\\mathcal{D}_v^{\\alpha}$ consists of $\\varrho$ such that there exists $x$ and $y$ such that \\[\\varrho(\\sigma_v)=c\\mtx{v}{x}{0}{1}\\text{ and } \\varrho(\\tau_v)=\\mtx{1}{y}{0}{1}.\\]Here $c$ is equal to $(\\kappa(\\sigma_v)\/v)^{\\frac{1}{2}}$.\n \\end{Remark}\n\\end{Def}\nWe shall denote by the kernel of $\\alpha$ restricted to $\\mathfrak{t}$ by $\\mathfrak{t}_{\\alpha}$. Since the action of $\\operatorname{G}_v$ on $\\operatorname{Ad}^0\\bar{\\rho}$ is trivial, \n\\[H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})=\\text{Hom}(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\] Let $\\mathcal{P}_v^{\\alpha}$ be the subspace of $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ consisting of $\\phi$ such that\n\\[\\phi(\\sigma_v)\\in \\mathfrak{t}_{\\alpha}+\\text{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha})\\]\n\\[\\phi(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}.\\]\nLet $\\Phi^{\\alpha}$ denote the subset of roots $\\beta\\in \\Phi$ such that $[(\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}, (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}]\\neq 0$. Recall that $X_{\\alpha}$ is a choice of root vector for $\\alpha$.\n\n \n\\begin{Def}\\label{defconditions}\n\\begin{enumerate}\n \\item Let $v$ be a trivial prime which is unramified mod $p^2$ in our lifting argument. Set $\\alpha=2L_1$ and $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$ consist of deformations with a representative \\[\\varrho'=(\\operatorname{Id}+X_{-\\alpha})\\varrho (\\operatorname{Id}+X_{-\\alpha})^{-1}\\] where $\\varrho$ is a representative for a deformation in $\\mathcal{D}_v^{\\alpha}$ which satisfies further conditions. In accordance with \\cite[Definition 3.5]{FKP1}, we assume that the mod-$p^2$ reduction $\\varrho_2:=\\varrho\\mod{p^2}$ satisfies the following conditions:\n \\begin{enumerate}\n \\item $\\varrho_2$ is unramified, with $\\varrho_2(\\sigma_v)\\in \\mathcal{T}(\\text{W}(\\mathbb{F}_q)\/p^2)$,\n \\item for all $\\beta\\in \\Phi^{\\alpha}$, \n \\[\\beta(\\varrho_2(\\sigma_v))\\neq 1\\mod{p^2}.\\]\n \\end{enumerate} Let $\\mathcal{S}_{v}^{\\alpha}$ consist of $\\phi\\in H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that $\\phi(\\sigma_v)\\in \\bigoplus_{\\beta\\in \\Phi^{\\alpha}} (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}$ and $\\phi(\\tau_v)=0$. Let $\\mathcal{N}_v$ be specified by \\[\\mathcal{N}_v=\\mathcal{N}_v^{nr}:=(\\operatorname{Id}+X_{-\\alpha})(\\mathcal{P}_v^{\\alpha}+\\mathcal{S}_v^{\\alpha})(\\operatorname{Id}+X_{-\\alpha})^{-1}.\\]\n \\item \\label{defconditions2}Let $v$ be a trivial prime which will be ramified mod $p^2$ in our lifting argument. Let $\\alpha=-2L_1$ and $\\mathcal{C}_v=\\mathcal{C}_v^{ram}$ consist of deformations in $\\mathcal{D}_v^{\\alpha}$ with representative $\\varrho$ satisfying some additional conditions, which we specify. In accordance with \\cite[Definition 3.9]{FKP1}, assume that the mod-$p^2$ reduction $\\varrho_2$ satisfies the following conditions:\n \\begin{enumerate}\n \\item $\\varrho_2(\\tau_v)\\in u_{\\alpha}(py)$ where $y\\in \\text{W}(\\mathbb{F}_q)^{\\times}$, and $u_{\\alpha}: (\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}\\rightarrow \\op{GSp}_{2n}$ is the root group homomorphism over $\\op{W}(\\mathbb{F}_q)$.\n \\item For all $\\beta\\in \\Phi^{\\alpha}$, \n \\[\\beta(\\varrho_2(\\sigma_v))\\neq 1\\mod{p^2}.\\]\n \\end{enumerate}Let $\\mathcal{S}_v^{\\alpha}$ denote the space of cohomology classes specified in the proof of \\cite[Lemma 3.10]{FKP1}. Let $\\mathcal{N}_v=\\mathcal{N}_v^{ram}$ be defined by \\[\\mathcal{N}_v:=\\mathcal{P}_v^{\\alpha}+\\mathcal{S}_v^{\\alpha}.\\] \n\\end{enumerate}\n\\end{Def}\nThe following gives us a criterion for an element $f\\in H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ to not be contained in $\\mathcal{N}_v^{nr}$. This criterion is used in the proof of Proposition $\\ref{lastchebotarev}$.\n\\begin{Lemma}\\label{lemma55}\nLet $v$ be a trivial prime and $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$. Let $f\\in \\mathcal{N}_v$, express $f(\\sigma_v)=\\sum_{\\lambda\\in \\Phi} {a_{\\lambda}} X_{\\lambda}+\\sum_{i=1}^n a_i H_i$. Write $X_{-2L_1}=c e_{n+1,1}$ and $X_{2L_1}=d e_{1,n+1}$. We have that $a_{2L_1}= -(cd)^{-1} a_{1}$.\n\\end{Lemma}\n\\begin{proof}\nSet $g:=(\\op{Id} + X_{-2L_1})^{-1} f(\\op{Id} + X_{-2L_1})$ and express $g(\\sigma_v)=\\sum_{\\lambda\\in \\Phi} {b_{\\lambda}} X_{\\lambda}+\\sum_{i=1}^n b_i H_i$. Note that for $\\phi\\in \\mathcal{P}_v^{2L_1}$, \\[\\phi(\\sigma_v)\\in \\mathfrak{t}_{2L_1}+\\text{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{2L_1})\\] and hence has zero $H_1$-component. For $\\phi\\in \\mathcal{S}_v^{2L_1}$, we have that\n\\[\\phi(\\sigma_v)\\in \\bigoplus_{\\beta\\in \\Phi^{2L_1}} (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}.\\] We deduce that the $H_1$-component $b_1$ is equal to zero. We show that the $H_1$-component of $g(\\sigma_v)$ is equal to $a_1+cd a_{2L_1}$ from the relation $g(\\sigma_v)=(\\op{Id}+X_{-2L_1})^{-1}f(\\sigma_v) (\\op{Id}+X_{-2L_1})$. Note that $X_{-2L_1}^2=0$ and thus, $(\\op{Id}+X_{-2L_1})^{-1}=(\\op{Id}-X_{-2L_1})$. One has that\n\\[\\begin{split}g(\\sigma_v)=&(\\op{Id}-X_{-2L_1})f(\\sigma_v) (\\op{Id}+X_{-2L_1})\\\\=&(\\op{Id}-c e_{n+1,1})f(\\sigma_v) (\\op{Id}+c e_{n+1,1})\\\\\n=&f(\\sigma_v)+ c[f(\\sigma_v),e_{n+1,1}]-c^2e_{n+1,1}f(\\sigma_v) e_{n+1,1}.\\end{split}\\]\nNote that \n\\[c^2e_{n+1,1}f(\\sigma_v) e_{n+1,1}=a_{2L_1}c^2 e_{n+1,1}X_{2L_1} e_{n+1,1}= a_{2L_1}c^2d e_{n+1,1}= a_{2L_1}cd X_{-2L_1},\\]and thus does not contribute to the $H_1$-component of $g(\\sigma_v)$. The contribution of \\[c[f(\\sigma_v), e_{n+1,1}]=\\sum_{\\lambda\\in \\Phi} {ca_{\\lambda}} [X_{\\lambda},e_{n+1,1}]+\\sum_{i=1}^n ca_i [H_i,e_{n+1,1}]\\] to the $H_1$-component of $g(\\sigma_v)$ is from the term\n\\[ca_{2L_1} [X_{2L_1},e_{n+1,1}]=ca_{2L_1} [d e_{1,n+1},e_{n+1,1}]=cd a_{2L_1} H_1.\\] Thus, we have shown that $b_1=a_1+cd a_{2L_1}$. Since, $b_1=0$, we deduce that $a_1= -cd a_{2L_1}$.\n\\end{proof}\n\\begin{Lemma}\\cite[Lemma 3.2, 3.6,3.10]{FKP1}\nLet $v$ be a trivial prime (for which either $\\mathcal{C}_v=\\mathcal{C}_v^{ram}$ or $\\mathcal{C}_v^{nr}$ is the chosen deformation condition) and $X\\in \\mathcal{N}_v$,\n\\begin{enumerate}\n \\item $\\dim \\mathcal{N}_v=\\dim\\operatorname{Ad}^0\\bar{\\rho}=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$.\n \\item Let $m\\geq 3$ and $\\rho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$, then\n\\[(\\operatorname{Id}+p^{n-1}X)\\rho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^{m}).\\]\n\\item The deformation functor $\\mathcal{C}_v$ is liftable.\n\\end{enumerate}\n\\end{Lemma}\n\\par Prior to lifting $\\bar{\\rho}$ to characteristic zero, we show that $\\bar{\\rho}$ lifts to $\\rho_2$ after increasing the set of ramification from $S$ to $S\\cup X_1$. One may choose a continuous lift $\\tau$ of $\\bar{\\rho}$ as depicted\n\\[ \\begin{tikzpicture}[node distance = 2.6 cm, auto]\n \\node(G) at (0,0) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X_1}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$};\n \\node (B) at (3,2){$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\tau$} (B);\n \\end{tikzpicture}\\]such that the composite $\\nu\\circ \\tau=\\psi \\mod{p^2}$.\n The obstruction class \\[\\mathcal{O}(\\bar{\\rho})_{\\restriction S\\cup X_1}\\in H^2(\\op{G}_{S\\cup X_1}, \\operatorname{Ad}^0\\bar{\\rho})\\] is represented by the $2$-cocycle\n \\[(g_1,g_2)\\mapsto \\tau(g_1 g_2)\\tau(g_2)^{-1}\\tau(g_1)^{-1}.\\]\n \n The residual representation $\\bar{\\rho}$ lifts to a representation $\\rho_2$ ramified only at primes in $S\\cup X_1$ if and only if this obstruction is zero. For $v\\in S$, the local representation $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$ which is a liftable deformation condition (by assumption) and thus lifts to mod $p^2$. The residual representation $\\bar{\\rho}$ is unramified at each prime $v\\in X_1$ and thus it is easy to see that $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ lifts to mod $p^2$ for $v\\in X_1$. As a consequence, $\\mathcal{O}(\\bar{\\rho})_{\\restriction S\\cup X_1}$ is contained in $\\Sh^2_{S\\cup X_1}(\\operatorname{Ad}^0\\bar{\\rho})$. We will show that a set of finitely many trivial primes $X_1$ can be chosen so that \\[\\Sh^2_{S\\cup X_1}(\\operatorname{Ad}^0\\bar{\\rho})=0.\\]For such a choice of $X_1$, there is a deformation $\\rho_2$\n \\[ \\begin{tikzpicture}[node distance = 2.8 cm, auto]\n \\node(G) at (0,0) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X_1}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$.};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\rho_2$} (B);\n \\end{tikzpicture}\\]\n \n\\begin{Prop}\\label{Shavanishing}\nLet $\\mathscr{M}$ denote the finite set of $\\operatorname{G}_{\\mathbb{Q}}$-modules defined by \\[\\begin{split}\\mathscr{M}:=&\\{(\\operatorname{Ad}^0\\bar{\\rho})\/(\\operatorname{Ad}^0\\bar{\\rho})_k, \\mid -2n+1\\leq k \\leq 2n\\}\\\\&\\cup \\{(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}, \\mid -2n+1\\leq k \\leq 2n\\}.\\\\\n\\end{split}\\]\nThere is a finite set $T\\supset S$ such that $T\\backslash S$ consists of only trivial primes such that for all $M\\in \\mathscr{M}$,\n\\begin{equation}\\label{equationTminusS}\n\\ker \\{H^1(\\op{G}_{\\mathbb{Q},T},M)\\rightarrow \\bigoplus_{w\\in T\\backslash S} H^1(\\op{G}_w, M)\\}=0\n\\end{equation}\nand so in particular,\n\\begin{equation*}\n\\Sh_T^1(M)=0.\n\\end{equation*}\n\\end{Prop}\n\\begin{proof}\nWe show that $T$ can be chosen for which \n\\begin{equation*}\n\\Sh_T^1(\\operatorname{Ad}^0\\bar{\\rho}^*)=0,\n\\end{equation*}\nthe argument for any $M\\in \\mathscr{M}$ is identical. For $0\\neq \\psi\\in H^1(\\operatorname{G}_{\\mathbb{Q},S}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, let $K_{\\psi}\\supset \\mathbb{Q}(\\operatorname{Ad}^0\\bar{\\rho}^*)$ be the field extension cut out by $\\psi$. By Lemma $\\ref{l4}$, the extension $K_{\\psi}$ is not equal to $\\mathbb{Q}(\\operatorname{Ad}^0\\bar{\\rho}^*)$. The extension $K(\\mu_{p^2})$ is linearly disjoint with $K_{\\psi}$ over $K$. By Lemma $\\ref{22Dec5}$, $K(\\mu_{p^2})$ is not contained in $K$ and $K(\\mu_{p^2})\\cap K_{\\psi}=K$. As a result, there is a nonempty Chebotarev class of primes which split in $K$ and are non-split in $K_{\\psi}$ and $K(\\mu_{p^2})$. If $v$ is such a prime, it must be a trivial prime since it splits in $K$ and is non-split in $\\mathbb{Q}(\\mu_{p^2})$. On the other hand, since $v$ is non-split in $K_{\\psi}$, deduce that $\\psi_{\\restriction \\op{G}_v}\\neq 0$. We may therefore choose a finite set of primes $T$ such that \n\\begin{itemize}\n\\item $T$ is finite,\n\\item $T\\backslash S$ consists of only trivial primes,\n\\item $\\ker \\{H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow \\bigoplus_{w\\in T\\backslash S} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)\\}=0$.\n\\end{itemize}\n\\end{proof}\nThe set of trivial primes $X_1$ is taken to be $T\\backslash S$.\n \n\\section{Lifting to mod $p^3$}\n\n\\par By Proposition $\\ref{Shavanishing}$, there is a finite set of primes $T$ containing $S$ such that $T\\backslash S$ consists of trivial primes and $\\Sh_{ T}^1(\\operatorname{Ad}^0\\bar{\\rho}^*)=0$. Let $X_1$ be the set of trivial primes $T\\backslash S$. At each prime $v\\in X_1$, let $\\mathcal{C}_v$ be the liftable deformation problem $\\mathcal{C}_v^{nr}$. By global duality, $\\Sh_{T}^2(\\operatorname{Ad}^0\\bar{\\rho})=0$ and thus the cohomological obstruction to lifting $\\bar{\\rho}$ to a representation $\\zeta_2$\n\\begin{equation}\\label{zeta2} \\begin{tikzpicture}[node distance = 2.2 cm, auto]\n \\node(G) at (0,0){$\\operatorname{G}_{\\mathbb{Q},T}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\zeta_2$} (B);\n \\end{tikzpicture}\\end{equation}\n vanishes. Here, $\\zeta_2$ is stipulated to have similitude character $\\kappa\\mod{p^2}$. Let $v\\in T$, recall that the set of $W(\\mathbb{F}_q)\/p^2$ lifts of $\\bar{\\rho}_{\\restriction \\op{G}_v}$ is an $H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$-torsor. Therefore there exists $z_v\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that the twist $(\\operatorname{Id}+z_v p) {\\zeta_2}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$. Further, for $v\\in X_1$, the class $z_v$ may is chosen so that this twist is unramified. We show that there is a set $W$ of at most two trivial primes such that on increasing the set $T$ to $Z=T\\cup W$ there exists a global cohomology class $h\\in H^1(\\op{G}_{\\mathbb{Q}, Z}, \\operatorname{Ad}^0\\bar{\\rho})$ such that\n \\begin{itemize}\n \n \\item $h_{\\restriction \\operatorname{G}_v}=z_v$ for $v\\in T$,\n \\item $(1+ph)\\zeta_{2}|_{\\op{G}_v}\\in \\mathcal{C}_v^{ram}$ for $v\\in W$.\n \\end{itemize} Further, letting $\\rho_2$ be the twist $\\rho_2=(\\operatorname{Id}+ph)\\zeta_2$, each local representation ${\\rho_2}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$ for $v\\in Z$. As a consequence, the obstruction class $\\mathcal{O}(\\rho_2)$ is in $ \\Sh_{Z}^2(\\operatorname{Ad}^0\\bar{\\rho})$. Since $Z$ contains $T$, the group $\\Sh_{Z}^2(\\operatorname{Ad}^0\\bar{\\rho})$ is zero. As a result, $\\rho_2$ must lift to $W(\\mathbb{F})\/p^3$. Assume that there is no such class $h$ for a set $W$ such that $\\# W\\leq 1$. It is shown that there is a pair of trivial primes $v_1,v_2\\notin T$ such that $W$ can be chosen to be equal to $\\{v_1,v_2\\}$. The set of trivial primes $X_2$ is then chosen to be $Z\\backslash S$. For $v\\in W$, choose $\\mathcal{C}_v$ to be equal to $\\mathcal{C}_v^{ram}$. In what follows, a Chebotarev class refers to a nonempty collection of primes defined by the application of the Chebotarev density theorem. Note that a Chebotarev class has positive Dirichlet density, and is in particular, infinite.\n \\begin{Prop}\\label{P1}\nLet $T$ be as in Proposition $\\ref{Shavanishing}$ and $\\psi$ be a nonzero element in $H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ and let $W\\subset H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ be a subspace not containing $\\psi$. Then, there exists a Chebotarev class of trivial primes $v$ such that \\[\\begin{split} &{\\psi}_{\\restriction \\operatorname{G}_v}\\neq 0\\\\\n&{\\beta}_{\\restriction \\operatorname{G}_v}=0 \\text{ for all } \\beta\\in W.\n\\end{split}\\]Moreover we may choose $v$ so that $v$ does not split completely in the $\\bar{\\chi} \\sigma_{2L_1}$-eigenspace of $\\operatorname{Gal}(K_{\\psi}\/K)$ when viewed as a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}^*$. \n\\end{Prop}\n\\begin{proof}\nLet $\\{\\psi_1,\\dots, \\psi_m\\}$ be a basis of $W$. Since $\\psi$ is not contained in the span of $W$, the classes $\\psi,\\psi_1,\\dots, \\psi_m$ are linearly independent. Extend $\\psi_1,\\dots, \\psi_m$ to $\\psi_1,\\dots, \\psi_r$, so that $\\psi,\\psi_1,\\dots, \\psi_r$ is a basis of $H^1(\\op{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$. Let $\\widetilde{W}$ be the span of $\\{\\psi_1,\\dots, \\psi_r\\}$. It suffices to prove the statement for $\\widetilde{W}$ in place of $W$, since $W$ is contained in $\\widetilde{W}$. Let $\\mathfrak{F}$ denote the composite $K_{\\psi_1} \\cdots K_{\\psi_r}$. Set $P:=\\operatorname{Gal}(K_{\\psi}\/K)$ and recall that $J_{\\psi}\\subset K_{\\psi}$ is the field fixed by $P_{\\bar{\\chi} \\sigma_{2L_1}}$. Lemma $\\ref{l4}$ asserts that $J_{\\psi}\\neq K_{\\psi}$. We will show that $\\mathfrak{F}\\cap K_{\\psi}\\subseteq J_{\\psi}$. First, we show how the result follows from this.\n\\par Set $\\mathfrak{L}:=\\mathfrak{F}\\cdot K_{\\psi}=K_{\\psi_1}\\cdots K_{\\psi_r}\\cdot K_{\\psi}$. We consider the following field diagram,\n\\begin{equation*}\n\\begin{tikzpicture}[node distance = 1.8cm, auto]\n \\node (Qmu) {$\\mathbb{Q}(\\mu_p).$};\n \\node (FK) [above of=Qmu, node distance= 1.25cm] {$\\mathfrak{F}\\cap K_{\\psi}$};\n \\node (J) [above of=FK, right of= FK, node distance= 0.9 cm] {$J_{\\psi}$};\n \\node (Kpsi) [above of=J, right of= J, node distance= 0.9 cm] {$K_{\\psi}$};\n \\node (F) [above of=FK, left of= FK] {\\small $\\mathfrak{F}$};\n \\node(P) [above of= Qmu, right of= Qmu] {$\\mathbb{Q}(\\mu_{p^2})$};\n \\node(FdotK)[above of= F, right of= F, node distance= 1.8cm] {$\\mathfrak{L}$};\n \\draw[-] (Qmu) to node {} (FK);\n \\draw[-] (Qmu) to node {} (P);\n \\draw[-] (FK) to node {} (J);\n \\draw[-] (FK) to node {} (F);\n \\draw[-] (F) to node {} (FdotK);\n \\draw[-] (J) to node {} (Kpsi);\n \\draw[-] (Kpsi) to node {} (FdotK);\n \\end{tikzpicture}\n \\end{equation*}\nBy Lemma $\\ref{22Dec5}$, the intersection $ K\\cap \\mathbb{Q}(\\mu_{p^2})=\\mathbb{Q}(\\mu_p) $. In fact, Lemma $\\ref{22Dec5}$ asserts that $\\mathfrak{F}\\cap\\mathbb{Q}(\\mu_{p^2})=\\mathbb{Q}(\\mu_p)$. Therefore there is a prime $v$ which is\n\\begin{enumerate}\n \\item split in $\\op{Gal}(\\mathfrak{F}\/\\mathbb{Q})$,\n \\item nonsplit in $\\op{Gal}(\\mathbb{Q}(\\mu_{p^2})\/\\mathbb{Q}(\\mu_p))$,\n \\item nonsplit in $\\op{Gal}(K_{\\psi}\/J_{\\psi})$.\n\\end{enumerate} Since $K=\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ is contained in $\\mathfrak{F}$, the prime $v$ is a trivial prime. Since $v$ splits in $\\op{Gal}(\\mathfrak{F}\/\\mathbb{Q})$, we have that $\\psi_{i\\restriction \\op{G}_v}=0$ for $i=1,\\dots, r$. Since $v$ does not split in $\\op{Gal}(K_{\\psi}\/K)$, we have that $\\psi_{\\restriction \\op{G}_v}\\neq 0$.\n\\par We begin by showing that $K_{\\psi}$ is not contained in $\\mathfrak{F}$. This is equivalent to the assertion that $\\mathfrak{L}$ is not equal to $\\mathfrak{F}$. Each of the classes $\\psi, \\psi_1,\\dots, \\psi_r$ is in the image of the inflation map \n\\[H^1(\\operatorname{Gal}(\\mathfrak{L}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\xrightarrow{\\op{inf}}H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*),\\]and hence the above map is an isomorphism. It follows that\n\\begin{equation*}\nh^1(\\operatorname{Gal}(\\mathfrak{L}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)=h^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)\\geq r+1.\\end{equation*}\nIt suffices to show that\n$h^1(\\operatorname{Gal}(\\mathfrak{F}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\leq r$. We show by induction on $i$ that\n\\begin{equation*}\nh^1(\\operatorname{Gal}( K_{\\psi_1}\\cdots K_{\\psi_i}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\leq i.\\end{equation*} Lemma $\\ref{l3}$ asserts that $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}^*)=0$ and hence by inflation-restriction,\n\\begin{equation*}\nH^1(\\operatorname{Gal}(K_{\\psi_1}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\simeq \\op{Hom}(P_1, \\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}.\n\\end{equation*}\nLemma $\\ref{y1}$ asserts that\n\\[\\dim \\op{Hom}(P_1,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1\\]and hence the case $i=1$ follows.\n\nFor the induction step, set $\\mathfrak{F}_i:=K_{\\psi_1}\\cdots K_{\\psi_{i}}$ and \\[P_{i}:=\\op{Gal}(\\mathfrak{F}_{i}\/\\mathfrak{F}_{i-1})\\simeq \\op{Gal}(K_{\\psi_{i}}\/K_{\\psi_{i}}\\cap \\mathfrak{F}_{i-1}).\\] Lemma $\\ref{y1}$ asserts that \\[\\dim \\op{Hom}(P_{i},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1\\]\nfrom which we see from inflation-restriction\n\\begin{equation*}\nh^1(\\operatorname{Gal}(\\mathfrak{F}_i\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\n \\leq h^1(\\operatorname{Gal}(\\mathfrak{F}_{i-1}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)+1.\n\\end{equation*}\n We conclude that $\\mathfrak{L}\\neq \\mathfrak{F}$ and thus we have deduced that $K_{\\psi}\\cap \\mathfrak{F}\\neq K_{\\psi}$. Set $Q:=\\op{Gal}(K_{\\psi}\/K_{\\psi}\\cap \\mathfrak{F})$, by Lemma $\\ref{mainin}$, \n \\begin{equation*}\n Q_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq P_{\\bar{\\chi}\\sigma_{2L_1}}.\n \\end{equation*}\nWe deduce that $K_{\\psi}\\cap \\mathfrak{F}$ is contained in $J_{\\psi}$. This completes the proof.\n\\end{proof}\n\\begin{Def}\nLet $\\mathcal{J}$ be a set of trivial primes that contains the set $S$ and $v\\notin \\mathcal{J}$ be a trivial prime. Denote by $\\Psi_{\\mathcal{J}}^k$ and $\\Psi_{\\mathcal{J},v}^k$ the maps defined by\n\\begin{equation*}\n\\Psi_{\\mathcal{J}}^k:H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{res_{\\mathcal{J}}}\\bigoplus_{w\\in \\mathcal{J}} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_k)\n\\end{equation*}\nand\n\\begin{equation*}\n\\Psi_{\\mathcal{J},v}^k: H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{res_\\mathcal{J}}\\bigoplus_{w\\in \\mathcal{J}} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_k).\n\\end{equation*}\nLet $\\tau_v$ be a generator of the maximal pro-$p$ quotient of the tame inertia at $v$, denote by\n\\begin{equation*}\n\\pi_{\\mathcal{J},v}^k: H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k\n\\end{equation*}\nthe evaluation map defined by\n\\[\\pi_{\\mathcal{J},v}^k(f):=f(\\tau_v).\\]\n\\end{Def}\n\\begin{Lemma}\\label{lemmaDec26}\nLet $T$ be a set of primes as in Proposition $\\ref{Shavanishing}$ that contains the set $S$ and $k$ an integer. Suppose $v\\notin T$ is a trivial prime with the property that for all $\\beta\\in H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)$, the restriction $\\beta_{\\restriction \\operatorname{G}_v}=0$. The following are exact:\n\\begin{equation}\\label{shortexact1}\n0\\rightarrow \\op{ker}\\Psi_{T}^k\\xrightarrow{inf} \\op{ker}\\Psi_{T,v}^k\\xrightarrow{\\pi_v^k} (\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow 0,\n\\end{equation} \n\\begin{equation}\\label{shortexact2}\n0\\rightarrow H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{inf} H^1(\\op{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{\\pi_v^k} (\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow 0.\n\\end{equation} \nFurther, the image of $\\Psi_T$ is equal to the image of $\\Psi_{T,v}$.\n\\end{Lemma}\n\n\\begin{proof}\nClearly the composite of the maps is zero and $\\eqref{shortexact1}$ is exact in the middle. Denote by $\\op{res}_v$ the restriction map: \\[\\op{res}_v:H^1(\\op{G}_{\\mathbb{Q}, T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\rightarrow H^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*).\\]By assumption, $H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)$ and $\\op{ker}\\op{res}_v$ are equal. By the local Euler characteristic formula and local duality, \\[h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)\\]\\[=h^2(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)=h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k.\\] By Wiles' Formula $\\eqref{wilesformula}$, \n\\[\\begin{split}\\dim \\op{ker}\\Psi_{T,v}^k=&\\dim \\op{ker}\\Psi_{T}^k+\\dim \\op{ker}\\op{res}_v-h^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\\\+&h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\\\\n=&\\dim \\op{ker}\\Psi_{T}^k+\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k\\\\\n\\end{split}\\]and the exactness of $\\eqref{shortexact1}$ follows. The exactness of $\\eqref{shortexact2}$ follows by the same arguments. Therefore, \n\\[\\begin{split}\\dim \\op{im} \\Psi_{T,v}=&h^1(\\op{G}_{\\mathbb{Q},T\\cup \\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k)-\\dim \\op{ker} \\Psi_{T,v}\\\\=&h^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_k)-\\dim \\op{ker} \\Psi_{T}=\\dim \\op{im} \\Psi_{T}.\\\\\\end{split}\\]\n\\end{proof}\nLet $M$ be an $\\mathbb{F}_q[\\operatorname{G}_{w}]$-module which is a finite dimensional $\\mathbb{F}_q$-vector space. The cup product induces the map\n\\[H^1(\\operatorname{G}_w, M)\\times H^1(\\operatorname{G}_w, M^*)\\rightarrow H^2(\\operatorname{G}_w, \\mathbb{F}_q(\\bar{\\chi}))\\xrightarrow{\\sim} \\mathbb{F}_q\\] taking $f_1\\in H^1(\\operatorname{G}_w, M)$ and $f_2\\in H^1(\\operatorname{G}_w, M^*)$ to $\\op{inv}_w(f_1\\cup f_2)\\in \\mathbb{F}_q$. Define the non-degenerate pairing \n\\[\\left(\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})\\right)\\times \\left(\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)\\right) \\rightarrow \\mathbb{F}_q\\]defined by\n$a\\cup b=\\sum_{w\\in T} \\op{inv}_w (a_w\\cup b_w)$. Denote by $\\op{Ann}((z_w)_{w\\in T})$ the annihilator of the tuple $(z_w)_{w\\in T}$. Recall that we assume that $(z_w)_{w\\in T}$ does not arise from a global class unramified outside $T$. In particular, the tuple $(z_w)_{w\\in T}$ is not zero, and as a result, $\\op{Ann}((z_w)_{w\\in T})$ is a codimension one subspace of $\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)$. Let $\\Psi_T$ and $\\Psi_T^*$ denote the restriction maps\n\\begin{equation*}\n\\begin{split}&\\Psi_{T}:H^1(\\operatorname{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho})\\rightarrow\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}),\\\\&\\Psi_{T}^*:H^1(\\operatorname{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*).\n\\end{split}\n\\end{equation*}\nFrom the exactness of the Poitou-Tate sequence \\cite[Theorem 8.6.14]{NW}, it follows that the images of $\\Psi_T$ and $\\Psi_T^*$ are exact annihilators of one another. Since it is assumed that $(z_w)_{w\\in T}$ is not in the image $\\Psi_T^*$, it follows that the image of $\\Psi_T$ is not contained in $\\op{Ann}((z_w)_{w\\in T})$. As a result, ${\\Psi_T^*}^{-1}(\\op{Ann}(z_w)_{w\\in T})$ has codimension one in $H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)$. Set $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ for the $\\mathbb{F}_q$ span of the root vector $X_{-2L_1}$.\n\\begin{Prop}\\label{P2}\nLet $T$ be as in Proposition $\\ref{Shavanishing}$. There exists a Chebotarev class $\\mathfrak{l}$ of trivial primes $v$ such that\n\\begin{enumerate}\n\\item\\label{22Decc1} $\\beta_{\\restriction \\operatorname{G}_v}=0$ for all $\\beta \\in H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_d^*)$ for $d\\geq -2n+2$,\n\\item\\label{22Decc2} there exists an $\\mathbb{F}_q$ basis $\\{\\psi, \\psi_1,\\dots, \\psi_r\\}$ of $H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ such that \n\\begin{itemize}\n\\item\\label{12}\n$\\{\\psi_1,\\dots, \\psi_r\\}$ is a basis of ${\\Psi_T^*}^{-1}(Ann(z_{w})_{w\\in T})$ \n\\item\n$\\psi_{\\restriction \\operatorname{G}_v}\\neq 0$ and ${\\psi_j}_{\\restriction \\operatorname{G}_v}=0$ for all $j\\geq 1$. \n\\end{itemize}\n\\end{enumerate}\nFurthermore, there is, for each $v\\in \\mathfrak{l}$, an element $h^{(v)}\\in H^1(\\operatorname{G}_{T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ such that \n\\[h^{(v)}|_{\\op{G}_w}=z_w\\] for all $w\\in T$ and \n\\begin{equation}\\label{equation64}\nh^{(v)}(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash \\{0\\}.\\end{equation}\n\\end{Prop}\n\\begin{proof}\nFirst, we analyze condition $\\eqref{22Decc1}$.\nRecall that $(\\operatorname{Ad}^0\\bar{\\rho})_d^*$ is the quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ by the Galois stable subspace $(\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp}$, see Definition $\\ref{perpdef}$. Its $\\mathbb{T}$-eigenspaces consist of $(\\operatorname{Ad}^0\\bar{\\rho})_{d,\\bar{\\chi}\\sigma_{\\lambda}^{-1}}^*$, where $\\lambda$ ranges through $\\Phi\\cup \\{1\\}$ with $\\op{ht}(\\lambda)\\geq d$. Condition $\\eqref{thc4}$ of Theorem $\\ref{main}$ asserts that $\\bar{\\chi}\\sigma_{-\\lambda}\\neq \\sigma_{1}$, i.e., $\\sigma_{\\lambda}\\neq \\bar{\\chi}$. Therefore, $(\\operatorname{Ad}^0\\bar{\\rho})_d^*$ contains no trivial eigenspace. Hence, the splitting conditions imposed by $\\eqref{22Decc1}$ are independent of the non-splitting condition in $\\mathbb{Q}(\\mu_{p^2})$ imposed by the fact that trivial primes are not $1\\mod{p^2}$. On the other hand, by Proposition $\\ref{P1}$, condition $\\eqref{22Decc2}$ can be satisfied by a Chebotarev class of trivial primes.\n\\par Next, we show that conditions $\\eqref{22Decc1}$ and $\\eqref{22Decc2}$ can be satisfied simultaneously. To show this, note that the condition requiring $\\psi_{\\restriction{\\op{G}_v}}\\neq 0$ is a non-splitting condition of $v$ in $\\op{Gal}(K_{\\psi}\/K)$. By Lemma $\\ref{l4}$, the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace for the $\\mathbb{T}$-action on $\\op{Gal}(K_{\\psi}\/K)$ is nontrivial. We shall require that $v$ does not split in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\op{Gal}(K_{\\psi}\/K)$. On the other hand, \\[(\\operatorname{Ad}^0\\bar{\\rho}^*)_d=\\bigoplus_{ \\op{ht}(\\lambda)\\geq d}(\\operatorname{Ad}^0\\bar{\\rho}^*)_{d,\\bar{\\chi}\\sigma_{\\lambda}^{-1}},\\]does not contain the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Note that the character $\\bar{\\chi}\\sigma_{2L_1}$ is not twist equivalent to any of the characters occurring in the $\\mathbb{T}$-eigenspace decomposition of $(\\operatorname{Ad}^0\\bar{\\rho}^*)_d$. As a result, it follows via an argument identical to that in proof of Lemma $\\ref{lemma416}$, that the non-splitting condition of $v$ in $\\op{Gal}(K_{\\psi}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}$ may be simultaneously satisfied along with the rest of the splitting conditions.\n\n\n\\par Let $v$ be a trivial prime which satisfies conditions $\\eqref{22Decc1}$ and $\\eqref{22Decc2}$ and moreover is non-split in $\\op{Gal}(K_{\\psi}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}$. Let $d\\geq -2n+2$, Lemma $\\ref{lemmaDec26}$ asserts that the image of\n\\begin{equation*}\n\\Psi_T^d:H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_d)\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_d)\n\\end{equation*}\nis the same as the image of\n\\begin{equation*}\n\\Psi_{T,v}^d:H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_d)\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_d).\n\\end{equation*}\nFor a trivial prime $v$ for which condition $\\eqref{22Decc2}$ is satisfied, it follows from an application of Wiles' formula $\\eqref{wilesformula}$ that the image of the map \n\\begin{equation*}\n\\Psi_{T,v}:H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})\n\\end{equation*}\nis greater than that of the map\n\\begin{equation*}\n\\Psi_T:H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}).\n\\end{equation*} We next deduce the existence of $h^{(v)}\\in H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ satisfying the specified properties. Since the image of $\\Psi_{T,v}$ is greater than the image of $\\Psi_T$, there is a class $g$ in $ H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ such that $\\Psi_{T,v}(g)\\notin \\text{Image}(\\Psi_T)$. Let \\[W_1:=\\text{Image}(\\Psi_T)+\\mathbb{F}_q\\cdot \\Psi_{T,v}(g)\\]\nand \n\\[W_2:=\\text{Image}(\\Psi_T)+\\mathbb{F}_q\\cdot (z_w)_{w\\in T}.\\]\nThe argument in \\cite[Proposition 34]{hamblenramakrishna} applies verbatim to imply that $W_1=W_2$ and so we deduce the existence of $h^{(v)}\\in H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ for which \\[h^{(v)}_{\\restriction \\op{G}_w}={z_w}_{\\restriction \\op{G}_w}\\] for all $w\\in T$. As we have observed,\n\\[\\text{Image}(\\Psi_{T}^{-2n+2})=\\text{Image}(\\Psi_{T,v}^{-2n+2})\\] since $h^{(v)}\\notin \\text{Image}(\\Psi_T)$ it follows that $h^{(v)}(\\tau_v)$ is not contained in $(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+2}$. Invoking Lemma $\\ref{lemmaDec26}$, we deduce that on adding a suitable linear combination of elements to $h^{(v)}$ from $\\ker \\Psi_{T,v}^d$ for $d> -2n+1$, we modify the class $h^{(v)}$ so that \\[h^{(v)}(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash \\{0\\}\\] as required. \n\\end{proof}\nFor $d\\in \\mathbb{Z}$, the natural inclusion $(\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp}\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ induces a natural map of cohomology groups $H^1(\\op{G}_{\\mathbb{Q}}, (\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp})\\rightarrow H^1(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$.\n\\begin{Lemma}\nLet $\\mathfrak{l}$ be the Chebotarev class of trivial primes in the Proposition $\\ref{P2}$. Let $\\{X_{\\lambda}^*\\}_{\\lambda\\in \\Phi}$ and $\\{H_1^*,\\dots, H_n^*\\}$ be as in $\\eqref{XHdual}$. There exists an $\\mathbb{F}_q$-independent set \\[\\{\\eta_{\\lambda}^{(v)}\\mid \\lambda \\in \\Phi\\}\\cup\\{\\ \\eta_{1}^{(v)}, \\dots,\\eta_{n}^{(v)}\\}\\] contained in $H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, satisfying the following properties:\n\\begin{enumerate}\n\\item\n$\\eta_{\\lambda}^{(v)}$ is in the image of the natural map \\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_{h+1}^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*),\\] where $h=\\op{ht}(\\lambda)$.\n\\item For $i=1,\\dots, n$, the cohomology class $\\eta_{i}^{(v)}$ \nis in the image of the natural map \\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_{1}^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\\]\n\\item \nFor $\\lambda \\in \\Phi$, we have that $\\eta_{\\lambda}^{(v)}(\\tau_v)= X_{\\lambda}^*$.\n\\item \nFor $i=1,\\dots, n$, we have that $\\eta_{i}^{(v)}(\\tau_v)=H_i^*$.\n\\item \nThe images of the elements $\\eta_{\\lambda}^{(v)}$ are a basis for the cokernel of the inflation map\n\\begin{equation*}\nH^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\\end{equation*}\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n The dual to $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}$ is $\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k$. Proposition $\\ref{Shavanishing}$ asserts that \\[\\Sh_T^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})=0\\] for all $k\\in \\mathbb{Z}$. Wiles' formula \\eqref{wilesformula} asserts that\n\\[\\begin{split}&h^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-\\dim \\Sh_{T\\cup\\{v\\}}^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})\\\\=& h^1(\\operatorname{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-\\dim \\Sh_{T}^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})\\\\+&h^1(\\op{G}_v, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-h^0(\\op{G}_v, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}).\\end{split}.\\] Also, by Proposition $\\ref{Shavanishing}$, we have that \\[\\Sh_T^1(\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0.\\] On applying the local Euler characteristic formula and Tate duality we have that \n\\[h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp}-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp})=h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}.\\]For the last equality, note that $\\bar{\\chi}_{\\restriction \\op{G}_v}=1$ since $v\\equiv 1\\mod{p}$ and that the action on $(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp}$ is trivial. It follows that \\[h^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=h^1(\\operatorname{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})+\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\] and the evaluation map at $\\tau_v$\n\\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\] induces a short exact sequence\n\\[0\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\rightarrow 0.\\] The assertion of the Lemma follows.\n\\end{proof}\nLet $v$ a trivial prime in the Chebotarev class $\\mathfrak{l}$ of Proposition $\\ref{P2}$. For $\\lambda\\in \\Phi$, denote by $K_{\\lambda}^{(v)}:=K_{\\eta_{\\lambda}^{(v)}}$ and for $i=1,\\dots, n$, set $K_i^{(v)}:=K_{\\eta_{i}^{(v)}}$. Let $J_i^{(v)}\\subsetneq K_i^{(v)}$ and $J_{\\lambda}^{(v)}\\subsetneq K_{\\lambda}^{(v)}$ denote $J_{\\eta_{i}^{(v)}}$ and $J_{\\eta_{\\lambda}^{(v)}}$ respectively. If $E=K_i^{(v)}$ (resp. $K_{\\lambda}^{(v)}$), denote by $J_E$ the sub-extension $J_{i}^{(v)}$ (resp. $J_{\\lambda}^{(v)}$). Let $\\mathcal{F}^{(v)}$ denote the collection of fields consisting of $K_i^{(v)}$ for $i=1,\\dots, n$ and $K_{\\lambda}^{(v)}$ for $\\lambda\\in \\Phi$. The Chebotarev class $\\mathfrak{l}$ from Proposition $\\ref{P2}$ is defined by Chebotarev classes in a collection of fields $\\mathcal{F}_{\\mathfrak{l}}$. More specifically, $\\mathcal{F}_{\\mathfrak{l}}$ is the collection of fields:\n\\begin{itemize}\n \\item $K_{\\psi}, K_{\\psi_1},\\dots, K_{\\psi_r}$ from Proposition $\\ref{P2}$,\n \\item $K_{\\beta}$ as $\\beta$ runs through all cohomology classes $H^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_d)$, where $d\\geq -2n+2$,\n \\item $K(\\zeta_2)$ (with $\\zeta_2$ defined at the start of the section),\n \\item $K(\\mu_{p^2})$.\n\\end{itemize}For $v\\in \\mathfrak{l}$ from Proposition $\\ref{P2}$, recall that $L_{h^{(v)}}$ is the field extension of $L$ cut out by \\[h^{(v)}_{\\restriction \\op{G}_L}:\\op{G}_L\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}.\\] Associate to a set of trivial primes $A=\\{v_1,\\dots, v_k\\}$ in $\\mathfrak{l}$, \\[\\mathcal{F}_A:=\\cup_{i=1}^k \\mathcal{F}^{(v_i)}\\text{, and } \\mathcal{L}_A:=\\{L_{h^{(v_1)}},\\dots, L_{h^{(v_k)}}\\}.\\]\n\\begin{Lemma}\\label{eigenspaceeta}\nLet $A=\\{v_1,\\dots, v_k\\}\\subset \\mathfrak{l}$.\n\\begin{enumerate}\n \\item\\label{66c1} Let $F_1$ be a field in the collection $\\mathcal{F}_A$ and $F_2$ be the composite of all the other fields in $ \\mathcal{F}_A\\cup\\mathcal{L}_A\\cup \\mathcal{F}_{\\mathfrak{l}}$. Then $F_1$ is not contained in $F_2$. Moreover, the intersection $F_1\\cap F_2$ is contained in $J_{F_1}$.\n \\item\\label{66c2} Let $M_1$ be a field in the collection $\\mathcal{L}_A$ and $M_2$ denote the composite of all the other fields in $\\mathcal{F}_A\\cup \\mathcal{L}_A\\cup \\mathcal{F}_{\\mathfrak{l}}$. The intersection $M_1\\cap M_2=L$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nPart $\\eqref{66c1}$ is obtained from an application of Proposition $\\ref{414}$, as we now explain. In accordance with the statement of Proposition $\\ref{414}$, we define a sequence of linearly independent classes \\[\\theta_0,\\dots, \\theta_t\\in H^1(\\op{G}_{\\mathbb{Q},T\\cup A},\\operatorname{Ad}^0\\bar{\\rho}^*),\\] and a sequence of fields $\\mathbb{L}_1, \\dots, \\mathbb{L}_b$, each of which is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\par Consider the classes $\\eta_i^{(v_j)}$ and $\\eta_{\\lambda}^{(v_j)}$ as $i=1,\\dots, n$, $\\lambda\\in \\Phi$ and $j=1,\\dots, k$. Enumerate these classes by $\\theta_0,\\dots, \\theta_a$, so that $\\theta_0$ is the cohomology class specified in the description of $F_1$, i.e. $F_1=K_{\\theta_0}$. The number $a$ is equal to $(nk\\# \\Phi)-1$ and $\\mathcal{F}_A=\\{K_{\\theta_0}, K_{\\theta_1},\\dots, K_{\\theta_a}\\}$. Let $\\theta_{a+1},\\dots, \\theta_{t}$ be a basis of $H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)$. Recall that for $\\lambda \\in \\Phi$, we have that $\\eta_{\\lambda}^{(v_j)}(\\tau_{v_j})= X_{\\lambda}^*$, and for $i=1,\\dots, n$, we have that $\\eta_{i}^{(v_j)}(\\tau_{v_j})=H_i^*$. The classes $\\theta_{a+1},\\dots, \\theta_{t}$ are unramified at each of the primes $v_j\\in A$. It is thus, easy to see that $\\theta_0,\\dots, \\theta_{t}$ are linearly independent. Let $\\mathbb{L}_1,\\dots, \\mathbb{L}_l$ be an enumeration for the fields $K_{\\beta}$, as $\\beta$ runs through all cohomology classes $H^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_d)$ for $d\\geq -2n+2$. Let $\\mathbb{L}_{l+1}$ be the field $K(\\zeta_2)$ and $\\mathbb{L}_{l+2}$ the field $K(\\mu_{p^2})$. The collection of fields $\\mathcal{F}_{\\mathfrak{l}}$ consists of $K_{\\theta_{i}}$ for $i=a+1,\\dots, t$ and $\\mathbb{L}_i$ for $i=1,\\dots, l+2$. Next, we have to account for the fields in $\\mathcal{L}_A$. Let $\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_b$ be an enumeration of the fields $L_{h^{(v_1)}}, \\dots, L_{h^{(v_k)}}$. Thus, the collection of fields $\\mathcal{L}_A$ is $\\{\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_b\\}$. In order to apply Proposition $\\ref{414}$, it suffices to show that each of the fields $\\mathbb{L}_1,\\dots, \\mathbb{L}_{b}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$. Note that:\n\\begin{itemize}\n \\item by Lemma $\\ref{415}$, part $\\eqref{415c1}$, each of the fields $\\mathbb{L}_{1},\\dots, \\mathbb{L}_l$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item by part $\\eqref{415c2}$, $\\mathbb{L}_{l+2}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item by part $\\eqref{415c3}$, $\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_{b}$ are unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item and by part $\\eqref{415c4}$, $\\mathbb{L}_{l+1}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\end{itemize}\n\n By Proposition $\\ref{414}$, $F_1$ is not contained in $F_2$ and it follows from Lemma $\\ref{mainin}$ that $F_1\\cap F_2\\subseteq J_{F_1}$.\n\\par Assume without loss of generality that $M_1=L_{h^{(v_1)}}$. Recall that by \\eqref{equation64}, we have that \\[h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}.\\]As a result, $v_1$ is ramified in the $\\sigma_{-2L_1}$-eigenspace of $\\op{Gal}(M_1\/L)$. On the other hand, $v_1$ is unramified in each of the field extensions in $\\mathcal{F}_{\\mathfrak{l}}$ and $\\mathcal{L}_A\\backslash \\{M_1\\}$. Since the classes $\\eta_i^{(v_j)}$ and $\\eta_{\\lambda}^{(v_j)}$ are valued in $\\operatorname{Ad}^0\\bar{\\rho}^*$, there is no $\\sigma_{-2L_1}$-eigenspace for the action of $\\mathbb{T}$ on $\\op{Gal}(K_{\\theta_i}\/K)$ for $K_{\\theta_i}\\in \\mathcal{F}_A$. As a result, $v_1$ is unramified in the $\\sigma_{-2L_1}$-eigenspace of $\\op{Gal}(M_2\/L)$. Therefore, $M_1\\not\\subseteq M_2$. Identify $Q:=\\operatorname{Gal}(M_1\/M_1\\cap M_2)$ with a subgroup of $h^{(v_1)}(\\op{G}_L)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. By Lemma $\\ref{fullrankLemma}$ it suffices to show that $Q_{-2L_1} \\neq 0$. Since $v_1$ is unramified in $M_2$, the image of $\\tau_{v_1}$ in $\\op{Gal}(M_1\/L)$ lies in $\\op{Gal}(M_1\/M_1\\cap M_2)$. Since $h^{(v_1)}(\\tau_{v_1})$ is in $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}$, we deduce that $Q_{-2L_1} \\neq 0$. The assertion $\\eqref{66c2}$ follows.\n\\end{proof}\n\\begin{Lemma}\nLet $v\\in \\mathfrak{l}$ and $h^{(v)}$ be as in Proposition $\\ref{P2}$. Then the $\\operatorname{Gal}(L_{h^{(v)}}\/L)\\simeq \\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{Lemma}\\label{hvLemma}\n\\begin{proof}\nLet $Q:=\\operatorname{Gal}(L_{h^{(v)}}\/L)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. Since $Q_{-2L_1} \\neq 0$, the assertion follows from Lemma $\\ref{fullrankLemma}$.\n\\end{proof}\n\\begin{Prop}\\label{lifttorho3} For a pair $(v_1,v_2)$ of trivial primes in $\\mathfrak{l}$ in Proposition $\\ref{P2}$ set $h=-h^{(v_1)}+2h^{(v_2)}$ and $\\rho_2:=(I+ph)\\zeta_2$. There is a pair $(v_1,v_2)$ such that $\\rho_{2\\restriction \\op{G}_w}\\in \\mathcal{C}_w$ for all $w\\in T$ and $\\rho_{2\\restriction \\op{G}_{v_i}}\\in \\mathcal{C}_{v_i}^{ram}$ for $i=1,2$.\n\\end{Prop}\n\\begin{proof}\nFor $i=1,2,$ we set $\\mathcal{C}_{v_i}:=\\mathcal{C}_{v_i}^{ram}$. Note that $h_{\\restriction \\op{G}_w}=z_w$\nfor all $w\\in T$ and hence $\\rho_{2\\restriction \\op{G}_w}\\in \\mathcal{C}_w$ for all $w\\in T$. This is not the case at the primes $v_1$ and $v_2$. We show that one may indeed find a pair $(v_1,v_2)\\in \\mathfrak{l}\\times \\mathfrak{l}$ so that $(I+pz_{v_i})\\zeta_2\\in \\mathcal{C}_{v_i}^{ram}$ for $i=1,2$. Consider for $v\\in \\mathfrak{l}$, the pair of elements $(\\zeta_2(\\sigma_v),h^{(v)}(\\sigma_v))$, and let $A=(A_1,A_2)$ be the pair of matrices which occurs most frequently, that is, with maximal upper density. The choice of $A$ is not necessarily unique. Let $\\mathfrak{l}_1=\\{v\\in \\mathfrak{l}\\mid \\zeta_2(\\sigma_v)=A_1,h^{(v)}(\\sigma_v)=A_2\\}$. Since there are finitely many choices for $A$, the set of primes $\\mathfrak{l}_1$ has positive upper-density. Since $h(\\tau_{v_i})\\in(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ and $\\zeta_2$ is unramified at $v_i$, we have that \\[(\\operatorname{Id}+ph(\\tau_{v_i}))\\zeta_2(\\tau_{v_i})=(\\operatorname{Id}+ph(\\tau_{v_i}))\\in \\op{U}_{-2L_1}.\\] Furthermore, since $h(\\tau_{v_i})\\neq 0$, the additional condition on $(\\operatorname{Id}+ph(\\tau_{v_i}))\\zeta_2(\\tau_{v_i})$ (see Definition $\\ref{defconditions}$ part $\\eqref{defconditions2}$) is satisfied. Since $\\zeta_2(\\sigma_v)$ is fixed throughout $\\mathfrak{l}_1$, there are (not necessarily unique) matrices $C_i$ such that if $h(\\sigma_{v_i})=C_i$, we will have \n$(\\operatorname{Id}+ph){\\zeta_2}_{\\restriction \\operatorname{G}_{v_i}} \\in \\mathcal{C}_{v_i}$ for $i=1,2$. The values $h^{(v_i)}(\\sigma_{v_j})$ are represented in the table below:\n\\begin{center}\n\\begin{tabular}{c|c|c } \n & $\\sigma_{v_1}$ & $\\sigma_{v_2}$ \\\\ [0.5 ex]\n \\hline\n $h^{(v_1)}$ & $A_2$ & $R$ \\\\\n \\hline\n $h^{(v_2)}$ & $E$ & $A_2$. \\\\ \n\\end{tabular}\n\\end{center}\nWe need $E=(A_2+C_1)\/2$ and $R=2A_2-C_2$. Note that for an arbitrary pair $(v_1,v_2)\\in \\mathfrak{l}_1\\times \\mathfrak{l}_1$, this need not be the case. What follows is a recipe for producing a pair $(v_1,v_2)$ such that $E=(A_2+C_1)\/2$ and $R=2A_2-C_2$.\n\\par For $v\\in \\mathfrak{l}_1$, let $\\delta^{(v)}\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)$ be the cohomology class given by $\\delta^{(v)}(\\sigma_v)=X_{-2L_1}^*$ and $\\delta^{(v)}(\\tau_v)=0$. Let $y$\nbe the element that occurs most frequently among the elements $\\operatorname{inv}_v (\\delta^{(v)} \\cup h^{(v)})$ among primes $v$ of $\\mathfrak{l}_1$. Set \\[\\mathfrak{l}_2 =\\{\nv\\in \\mathfrak{l}_1 \\mid \\operatorname{inv}_v (\\delta^{(v)} \\cup h^{(v)})= y\\},\\] $\\mathfrak{l}_2$ has positive\nupper density. Suppose we first choose $v_1\\in \\mathfrak{l}_2$. Recall that $h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$. By Lemma $\\ref{fullrankLemma}$, the class $h^{(v_1)}$ has full rank, i.e. $h^{(v_1)}(\\op{G}_K)=\\operatorname{Ad}^0\\bar{\\rho}$. In particular, $2A_2-C_2$ is contained in $h^{(v_1)}(\\op{G}_K)$. Choosing $v_2$ such that $h^{(v_1)}(\\sigma_{v_2})=2A_2-C_2$ is a Chebotarev condition on the splitting of $v_2$ in $L_{h^{(v_1)}}$. We show that $h^{(v_2)}(\\sigma_{v_1})$ is determined by how $v_2$ splits in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace each of the fields in $\\mathcal{F}^{(v_1)}$. Since $h^{(v_2)}$ is unramified at $v_1$, the values $\\eta_{\\lambda}^{(v_1)}(\\tau_{v_1})$ and $h^{(v_2)}(\\sigma_{v_1})$ determine $(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})_{\\restriction \\op{G}_{v_1}}$. Express $h^{(v_2)}(\\sigma_{v_1})=\\sum_{\\lambda} a_{\\lambda} X_{\\lambda}+\\sum_{i=1}^n a_{i} H_{i}$. As $\\eta_{\\lambda}^{(v_1)}(\\tau_{v_1})= X_{\\lambda}^*$, we see that $\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$ determines $a_{\\lambda}$. Likewise, $\\op{inv}_{v_1}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})$ determines $a_{i}$. For $v\\in \\mathfrak{l}$ and $\\lambda\\in \\Phi$, set $z_{\\lambda}^{(v)}$ to be equal to $\\op{inv}_v(\\eta_{\\lambda}^{(v)}\\cup h^{(v)})$. The global reciprocity law asserts that\n\\[\\sum_{w\\in T\\cup \\{v_1,v_2\\}} \\op{inv}_w(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})=0,\\text{ and }\\sum_{w\\in T\\cup \\{v_1\\}} \\op{inv}_w(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})=0.\\]Since $h^{(v_2)}_{\\restriction \\op{G}_w}=z_w=h^{(v_1)}_{\\restriction \\op{G}_w}$ for $w\\in T$, we deduce that\n\\begin{equation*}\n\\begin{split}\n\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)}) &=-\\sum_{w\\in T} \\op{inv}_{w}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=-\\sum_{w\\in T} \\op{inv}_{w}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=z_{\\lambda}^{(v_1)}-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)}).\n\\end{split}\n\\end{equation*}\nSince $z_{\\lambda}^{(v_1)}$ depends on $v_1$ which is fixed, the variance of the right hand side of the equation comes from the term $\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$. The specification of $h^{(v_2)}(\\sigma_{v_1})$ amounts to the specification of $\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$ for $\\lambda\\in \\Phi$ and $\\op{inv}_{v_1}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})$ for $i=1,\\dots,n$. Set $u_{\\lambda}$ to be $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $\\lambda \\in \\Phi$ and set $u_i$ to be $\\eta_{i}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $i=1,\\dots, n$. Since $h^{(v_2)}(\\tau_{v_2})$ is a multiple of $X_{-2L_1}$, we see that \n\\[\\begin{split}&\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})=\\op{inv}_{v_2}(u_{\\lambda}\\delta^{(v_2)}\\cup h^{(v_2)})\\\\\n&\\op{inv}_{v_2}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})=\\op{inv}_{v_2}(u_{i}\\delta^{(v_2)}\\cup h^{(v_2)}).\\\\\n\\end{split}\\] Moreover since $h^{(v_2)}(\\tau_{v_2})$ is non-zero, we can choose $b\\in \\mathbb{F}_q$ such that $\\op{inv}_{v_2}(b\\delta^{(v_2)}\\cup h^{(v_2)})$ takes on any desired value. Note that $\\op{inv}_{v_2}(\\delta^{(v_2)}\\cup h^{(v_2)})$ is set to equal $y$ for all $v_2\\in \\mathfrak{l}_2$, i.e., does not depend on the choice of $v_2\\in \\mathfrak{l}_2$. As a result, for $v_1\\in \\mathfrak{l}_2$, there exist values $\\{b_{\\lambda}\\}_{\\lambda\\in \\Phi}$ and $\\{b_i\\}_{i=1,\\dots, n}$ depending only on $v_1$ such that if $u_{\\lambda}=b_{\\lambda}$ for $\\lambda \\in \\Phi$ and $u_{i}=b_{i}$ for $i=1,\\dots, n$, then, \\[h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2.\\] The condition requiring $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$, is determined by $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $\\lambda \\in \\Phi$ and by $\\eta_{i}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $i=1,\\dots, n$. Note that $v_2$ is unramified in $K_{\\eta_{\\lambda}^{(v_1)}}$ and the value of $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ is determined by the projection of $\\sigma_{v_2}$ to the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/K)$, when viewed as a $\\mathbb{T}$-module. Recall that $J_{\\eta_{\\lambda}^{(v_1)}}$ is the subextension $K\\subseteq J_{\\eta_{\\lambda}^{(v_1)}}\\subsetneq K_{\\eta_{\\lambda}^{(v_1)}}$ such that \\[\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq \\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/J_{\\eta_{\\lambda}^{(v_1)}}).\\] Since $v_2$ is a trivial prime, it is split in $K$. One may indeed insist that $v_2$ is split in $J_{\\eta_{\\lambda}^{(v_1)}}$ and $\\sigma_{v_2}$ takes on the appropriate value in $\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/J_{\\eta_{\\lambda}^{(v_1)}})$ so that $u_{\\lambda}=b_{\\lambda}$. Hence, the condition $u_{\\lambda}=b_{\\lambda}$ is simply a condition on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $K_{\\eta_{\\lambda}^{(v_1)}}\/K$. Likewise, the condition $u_{i}=b_{i}$ is a condition on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $K_{\\eta_{i}^{(v_1)}}\/K$. To summarize, the condition requiring $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$, is equivalent to Chebotarev conditions on the splitting of $v_2$ in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspaces of the fields in $\\mathcal{F}^{(v_1)}$, in the sense made precise in the preceding discussion.\n\\par Suppose that for the choice of $v_1\\in \\mathfrak{l}_2$, there is a $v_2\\in \\mathfrak{l}_2$ for which the required conditions are satisfied:\n\\begin{enumerate}\n \\item the condition on the splitting of $v_2$ in $L_{h^{(v_1)}}$ which amounts to specifying $h^{(v_1)}(\\sigma_{v_2})$,\n \\item the condition on the splitting of $v_2$ in the fields $\\mathcal{F}^{(v_1)}$ which amounts to specifying $h^{(v_2)}(\\sigma_{v_1})$.\n\\end{enumerate}Then we are done. Note that $\\mathfrak{l}_2$ is not a Chebotarev condition, it has only been observed that $\\mathfrak{l}_2$ has positive upper density. Consider the case when there is no choice of $v_2\\in \\mathfrak{l}_2$ for which the above conditions are satisfied. Let $\\mathfrak{l}_{v_1}$ be the subset of $\\mathfrak{l}$ for which $(R, E)\\neq (2A_2-C_2, \\frac{(A_2 + C_1)}{2})$ for the choice of $v_1$. We have thus assumed that $\\mathfrak{l}_2\\subseteq \\mathfrak{l}_{v_1}$, it follows that the upper density $\\delta(\\mathfrak{l}_2)$ is less than or equal to the upper density $\\delta(\\mathfrak{l}_{v_1})$. \n\\par Set $\\mathcal{E}^{(v_1)}$ to be the composite of the field $L_{h^{(v_1)}}$ with the fields in $\\mathcal{F}^{(v_1)}$ and let $\\mathfrak{F}_{\\mathfrak{l}}$ be the composite of fields in $\\mathcal{F}_{\\mathfrak{l}}$. We show that there is an element $x\\in\\op{Gal}(\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}\/K)$ such that if $v_2$ is trivial prime such that the Frobenius at $v_2$ maps to $x$, then $v_2\\in \\mathfrak{l}$ and the conditions on $v_2$ are satisfied. Said differently, if $\\sigma_{v_2}=x$, then $v_2\\in \\mathfrak{l}\\backslash \\mathfrak{l}_{v_1}$. If $F_1$ is any of the fields in $\\mathcal{F}^{(v_1)}$ and $F_2$ is the composite of the other fields in $\\mathcal{F}^{(v_1)}\\cup \\mathcal{F}_{\\mathfrak{l}}$, Lemma $\\ref{eigenspaceeta}$ asserts that $F_1\\cap F_2\\subseteq J_{F_1}$. Lemma $\\ref{eigenspaceeta}$ asserts that $L_{h^{(v_1)}}$ is linearly disjoint over $L$ from the composite of all fields in $\\mathcal{F}^{(v_1)}\\cup \\mathcal{F}_{\\mathfrak{l}}$. To construct such an element $x$, enumerate the fields in $\\mathcal{F}^{(v_1)}=\\{E_1,\\dots, E_{k-1}\\}$ and set $E_{k}:=F_{h^{(v_1)}}$. Set $E_0:=\\mathfrak{F}_{\\mathfrak{l}}$ and let $\\mathcal{E}_j$ be the composite $E_0\\cdots E_j$, note that $\\mathcal{E}_{k}=\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}$. Consider the filtration\n\\[\\mathcal{E}_{k}\\supset \\mathcal{E}_{k-1}\\supset \\dots \\supset \\mathcal{E}_1\\supset \\mathcal{E}_0\\supset K.\\] Let $x_0\\in \\op{Gal}(\\mathcal{E}_0\/K)$ be an element defining $\\mathfrak{l}$. Note that $\\op{Gal}(\\mathcal{E}_1\/\\mathcal{E}_0)\\simeq \\op{Gal}(E_1\/E_1\\cap \\mathcal{E}_0)$ and the intersection $E_1\\cap \\mathcal{E}_0$ is contained in $J_{E_1}$. The condition on $E_1\/K$ is on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace $\\op{Gal}(E_1\/J_{E_1})$. Hence $x_0$ lifts to a suitable $x_1\\in \\op{Gal}(\\mathcal{E}_1\/K)$. Repeating the process, we see that $x_1$ lifts to $x_{k-1}\\in \\op{Gal}(\\mathcal{E}_{k-1}\/K)$ such that if $\\sigma_{v_2}=x_{k-1}$, then $v_2\\in \\mathfrak{l}$ and $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$. Since $E_k\\cap \\mathcal{E}_{k-1}=K$, it follows that $x_{k-1}$ can be lifted to $x_{k}\\in \\op{Gal}(\\mathcal{E}_{k}\/K)$ such that if $\\sigma_{v_2}=x$, then all conditions on $v_2$ are satisfied.\n\nAs a result, $\\delta(\\mathfrak{l}\\backslash \\mathfrak{l}_{v_1})\\geq \\frac{1}{[\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}:K]}$, and hence,\n\\[\\delta(\\mathfrak{l}_{v_1})\\leq \\left(1-\\frac{1}{[\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}:K]}\\right).\\]For $F\\in \\mathcal{F}^{(v_1)}$, the Galois group $\\op{Gal}(F\/K)$ may be identified with a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Hence $[F:K]\\leq q^{\\dim(\\operatorname{Ad}^0\\bar{\\rho})}$ for $F\\in \\mathcal{F}^{(v_1)}$ is a uniform bound independent of $v_1$. Similar reasoning shows that $[L^{h^{(v_1)}}:L]\\leq q^{\\dim (\\operatorname{Ad}^0\\bar{\\rho})} $. Setting $N:=(\\#\\Phi +n+1)\\cdot \\dim \\operatorname{Ad}^0\\bar{\\rho}$, deduce that \\[\\delta(\\mathfrak{l}_{v_1})\\leq 1-q^{-N}[\\mathfrak{F}_{\\mathfrak{l}}:K]^{-1}.\\]\n\\par Suppose that there is a sequence of $m$ primes $v_{1}^{(1)},\\dots, v_{1}^{(m)}\\in \\mathfrak{l}_2$, such that it is not possible to find a second prime $v_2$ for any of the primes $v_1^{(j)}$. In other words, $\\mathfrak{l}_2\\subseteq \\cap_{j=1}^m \\mathfrak{l}_{v_1^{(j)}}$. We show that the density of $\\cap_{j=1}^m \\mathfrak{l}_{v_1^{(j)}}$ approaches zero as $m$ approaches infinity. Since the upper density of $\\mathfrak{l}_2$ is positive, we will eventually find a pair $(v_1,v_2)$. For convenience of notation, set $w_j:=v_1^{(j)}$ and set $A=\\{w_1,\\dots, w_m\\}$. Fix $1\\leq j\\leq m$ and enumerate the fields $\\mathcal{F}^{(w_j)}=\\{E_1,\\dots, E_{k-1}\\}$ and set $E_k=F_{h^{(w_j)}}$. Denote by $\\mathfrak{E}_j:=\\mathfrak{F}_{\\mathfrak{l}}\\cdot \\mathcal{E}^{(w_1)}\\cdots\\mathcal{E}^{(w_j)}$ and let $C_j$ be the subset of $\\op{Gal}(\\mathfrak{E}_j\/K)$ defining the set $\\cap_{i=1}^j \\mathfrak{l}_{w_i}$. This means that $v_2\\in \\cap_{i=1}^j \\mathfrak{l}_{w_i}$ if and only if $\\sigma_{v_2}\\in C_j$. We show that any element $w\\in \\op{Gal}(\\mathfrak{E}_{j-1}\/K)$ lifts to an element $\\tilde{w}\\in\\op{Gal}(\\mathfrak{E}_{j}\/K)$ which is not in $C_j$. This is shown by filtering $\\mathfrak{E}_j\/\\mathfrak{E}_{j-1}$ by \n\\[\\mathfrak{E}_{j}=\\mathcal{E}_k\\supset \\mathcal{E}_{k-1}\\supset \\cdots \\mathcal{E}_1\\supset \\mathcal{E}_0=\\mathfrak{E}_{j-1},\\]\nwhere $\\mathcal{E}_l:=\\mathfrak{E}_{j-1}E_1\\cdots E_l$. The argument is identical to that provided before.\n\\par\nAs a result, \\[\\# C_j\\leq ([\\mathfrak{E}_j:\\mathfrak{E}_{j-1}] -1)\\# C_{j-1}.\\] Therefore,\n\\[\\begin{split}\\delta(\\cap_{i=1}^j \\mathfrak{l}_{w_i})=\\frac{\\# C_j}{[\\mathfrak{E}_j:K]}\\leq & \\left(1-\\frac{1}{[\\mathfrak{E}_j:\\mathfrak{E}_{j-1}]}\\right)\\frac{\\# C_{j-1}}{[\\mathfrak{E}_{j-1}:K]},\\\\\n\\leq & \\left(1-\\frac{1}{[\\mathcal{E}^{(w_j)}:K]}\\right)\\frac{\\# C_{j-1}}{[\\mathfrak{E}_{j-1}:K]},\\\\\n\\leq & (1-q^{-N}) \\delta(\\cap_{i=1}^{j-1} \\mathfrak{l}_{w_i}).\n\\end{split}\\]Therefore, $\\delta(\\cap_{i=1}^m \\mathfrak{l}_{w_i})\\leq (1-q^{-N})^{m-1} (1-q^{-N}[\\mathfrak{F}_{\\mathfrak{l}}:K]^{-1})$. Since $\\mathfrak{l}_2$ has positive upper density there is a large value of $m$ such that $\\mathfrak{l}_2$ is not contained in $\\cap_{i=1}^m \\mathfrak{l}_{w_i}$. This shows that a pair $(v_1,v_2)$ satisfying the required conditions does exist.\n\\end{proof}\n\n\\begin{Prop}\\label{bigimageprop}\nLet $\\rho_2$ be as in Proposition $\\ref{lifttorho3}$. The image of $\\rho_2$ is the principal congruence subgroup of $\\op{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$ of similitude character $1$.\n\\end{Prop}\n\\begin{proof}\nRecall that the similitude character $\\kappa$ is prescribed to equal $\\kappa_0\\chi^k$ where $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$ and $k$ is divisible by $p(p-1)$. Therefore, we have that $\\kappa\\equiv \\kappa_0\\mod{p^2}$, and as a result, elements in the principal congruence subgroup in the image of $\\rho_2$ necessarily have similitude character $1$. Therefore, $\\rho_2(\\op{G}_L)$ may be identified with a subspace of $\\operatorname{Ad}^0\\bar{\\rho}$. In greater detail, $\\rho_2(g)$ is identified with $\\frac{1}{p}(\\rho_2(g)-\\op{Id})$, for $g\\in \\op{G}_L=\\op{ker}\\bar{\\rho}$. It may be checked that $\\rho_2(\\op{G}_L)$ is a $\\op{G}$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ and that the natural $\\op{G}$-action on $\\op{Gal}(\\mathbb{Q}(\\rho_2)\/L)$ (induced by conjugation) coincides with the $\\op{G}$-action on $\\rho_2(\\op{G}_L)$ viewed as a submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Recall that $\\rho_2=(\\op{Id}+ph)\\zeta_2$, where $h$ is the cohomology class given by $-h^{(v_1)}+2h^{(v_2)}$. Since $\\bar{\\rho}$ is unramified at $v_1$, we have that $\\tau_{v_1}\\in \\op{G}_L$. The cohomology class $h^{(v_2)}$ is unramified at $v_1$, as is $\\zeta_2$. Therefore, we have that\n\\[\\rho_2(\\tau_{v_1})=(\\op{Id}+ph(\\tau_{v_1}))\\zeta_2(\\tau_{v_1})=(\\op{Id}-ph^{(v_1)}(\\tau_{v_1})).\\]\nRecall that by \\eqref{equation64}, we have that \\[h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}.\\]Therefore, $\\rho_2(\\op{G}_L)$ is identified with a Galois-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ which contains an element with non-zero $-2L_1$-component. From Lemma $\\ref{fullrankLemma}$, it is deduced that this module must be all of $\\operatorname{Ad}^0\\bar{\\rho}$. This completes the proof.\n\\end{proof}\n\\section{Annihiliating the dual-Selmer Group}\nLet $\\rho_3:\\operatorname{G}_{\\mathbb{Q},T\\cup \\{v_1,v_2\\}}\\rightarrow \\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^3)$ be the lift of $\\bar{\\rho}$ obtained from the application of Propositions $\\ref{lifttorho3}$ and $\\ref{bigimageprop}$. Recall that the Galois group $\\op{Gal}(K(\\rho_2)\/K)$ is identified with $\\operatorname{Ad}^0\\bar{\\rho}$. As a result, once it is shown that $\\rho_3$ lifts to a characteristic zero representation $\\rho$, it shall follow that $\\rho$ is irreducible. In showing that $\\rho_3$ can be lifted to characteristic zero, we enlarge the set of primes $Z=T\\cup \\{v_1,v_2\\}$ to a finite set of primes $Y$ such that $X:=Y\\backslash S$ consists only of trivial primes. For $i=1,2$ set $\\mathcal{C}_{v_i}=\\mathcal{C}_{v_i}^{ram}$ and for primes $v\\in X\\backslash \\{v_1,v_2\\}$, set $ \\mathcal{C}_v=\\mathcal{C}_v^{nr}$. We show that the dual-Selmer group $H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}^*)$ vanishes for a suitably chosen set of primes $Y$. For convenience of notation, denote by $\\mathscr{W}$ the Galois submodule $(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+2}$ of $\\operatorname{Ad}^0\\bar{\\rho}$ spanned by root spaces $(\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}$ for $\\beta\\neq -2L_1$. \n\\begin{Prop}\\label{lastchebotarev}\nLet $\\rho_3:\\operatorname{G}_{\\mathbb{Q},T\\cup \\{v_1,v_2\\}}\\rightarrow \\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^3)$ be the lift of $\\bar{\\rho}$ obtained from the application of Propositions $\\ref{lifttorho3}$ and $\\ref{bigimageprop}$. Let $Y$ be a finite set of primes which contains $Z=T\\cup \\{v_1,v_2\\}$ such that $Y\\backslash S$ consists of trivial primes. Suppose $f\\in H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})$ and $\\psi\\in H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ are nonzero classes. Then there exists a prime $v\\notin Y$ such that\n\\begin{enumerate}\n\\item \\label{71one} $v$ is a trivial prime,\n\\item \\label{71two}$\\rho_{3\\restriction \\op{G}_v}$ satisfies $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$, \n\\item \\label{71three} $f$ does not satisfy $\\mathcal{N}_v=\\mathcal{N}_v^{nr}$,\n\\item\\label{71four}$\\beta_{\\restriction \\operatorname{G}_v}=0$ for all $\\beta\\in H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)$,\n\\item \\label{71five} $\\psi_{\\restriction \\operatorname{G}_v}\\neq 0$ and one can extend $\\{\\psi\\}$ to a basis $\\psi_1=\\psi,\\psi_2,\\dots, \\psi_k$ of $H^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ such that $\\psi_{i \\restriction \\operatorname{G}_v}=0$ for $i>1$.\n\\end{enumerate}\n\\end{Prop}\n\\begin{proof}\nEach condition is a union of Chebotarev conditions on a number of finite extensions $J$ of $K$. Each of the extensions $J$ are Galois over $\\mathbb{Q}$ with $\\op{Gal}(J\/K)$ an $\\mathbb{F}_p$-vector space. Let $g\\in \\op{G}'$ and $x\\in \\op{Gal}(J\/K)$, define, $g\\cdot x:=\\tilde{g} x \\tilde{g}^{-1}$ where $\\tilde{g}$ is a lift of $g$ to $\\op{Gal}(J\/\\mathbb{Q})$. This gives $\\op{Gal}(J\/K)$ the structure of an $\\mathbb{F}_p[\\op{G}']$-module. For each condition, we list the choices for $J$ below as well as characters for the $\\mathbb{T}$-action on $\\op{Gal}(J\/K)$:\n\\begin{center}\n\\begin{tabular}{c|c|c } \n \\text{Condition} & $J$ & $\\text{Eigenspaces of } \\operatorname{Gal}(J\/K)$ \\\\ [1 ex]\n \\hline\n $(1)$ & $K(\\mu_{p^2})$ & $1$ \\\\\n \\hline\n $(2)$ & $K(\\rho_2)$ & $1,\\{\\sigma_{\\lambda}\\}_{\\lambda\\in \\Phi}$ \\\\\n \\hline\n $(3)$ & $K_f$ & $1,\\{\\sigma_{\\lambda}\\}_{\\lambda\\in \\Phi}$ \\\\\n \\hline\n $(4)$ & $K_{\\beta}$ \\text{ for } $\\beta \\in H^1(\\op{G}_{\\mathbb{Q},Y},\\mathscr{W}^*)$& $\\bar{\\chi},\\{\\bar{\\chi}\\sigma_{\\lambda}^{-1}|\\lambda\\neq -2L_1\\}$\\\\\n \\hline\n $(5)$ & $K_{\\psi_i} $ & $\\bar{\\chi},\\{\\bar{\\chi}\\sigma_{\\lambda}^{-1}\\}$.\\\\\n\\end{tabular}\n\\end{center}\nWe show that these conditions may be simultaneously satisfied. First, we show that each of the conditions is a nonempty Chebotarev condition (or a union of finitely many Chebotarev conditions). It is clear that condition $\\eqref{71one}$ and $\\eqref{71two}$ are nonempty Chebotarev conditions. Lemma $\\ref{lemma55}$ gives a criterion for the element $f$ to not be in the space $\\mathcal{N}_v$. In accordance with Lemma $\\ref{lemma55}$, write $X_{-2L_1}=c e_{n+1,1}$ and $X_{2L_1}=d e_{1,n+1}$. Since $f$ is non-zero, $f(\\op{G}_L)$ is a non-zero Galois-stable submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Hence, by Lemma $\\ref{mainin}$, contains $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$. Therefore, the image of $f_{\\restriction \\op{G}_L}$ contains an element \n\\[f(g)=\\sum_{\\lambda\\in \\Phi} a_{\\lambda} X_{\\lambda} +\\sum_{i=1}^n a_i H_i\\]such that $a_{2L_1}\\neq -(cd)^{-1} a_1$. As a result, condition $\\eqref{71three}$ is a union of finitely many nonempty Chebotarev conditions. Condition $\\eqref{71four}$ requires that the prime splits in the composite of the fields $K_{\\beta}$. That condition $\\eqref{71five}$ is a nonempty Chebotarev condition follows from Proposition $\\ref{P2}$.\n\nNext we examine the independence of these conditions. It follows from Lemma $\\ref{lemma416}$ that the composite of the fields defining the first three conditions is linearly disjoint over $K$ from the composite of the fields defining the last two conditions. As a result, the conditions may be treated separately from the last two. It follows from Proposition $\\ref{P2}$ that the conditions $\\eqref{71four}$ and $\\eqref{71five}$ are compatible with each other. Therefore, it remains to show that $\\eqref{71one}$,$\\eqref{71two}$ and $\\eqref{71three}$ may be simultaneously satisfied. We begin with the independence of $\\eqref{71one}$ and $\\eqref{71two}$. Proposition $\\ref{bigimageprop}$ asserts that $\\operatorname{Gal}(K(\\rho_2)\/K)= \\operatorname{Ad}^0\\bar{\\rho}$. Suppose that $Q$ is a proper $\\op{G}'$-stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}$. Lemma $\\ref{Pdecomposition}$ asserts that $Q$ decomposes into $\\mathbb{T}$-eigenspaces $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$ and Lemma $\\ref{fullrankLemma}$ asserts that the eigenspace $Q_{-2L_1}:=Q_{\\sigma_{-2L_1}}$ must be trivial. Hence the quotient $\\operatorname{Ad}^0\\bar{\\rho}\/Q$ must have a non-zero $\\sigma_{-2L_1}$-eigenspace. It follows that there is no proper Galois stable subgroup $Q$ of $\\operatorname{Ad}^0\\bar{\\rho}$ such that $\\operatorname{Ad}^0\\bar{\\rho}\/Q$ is has trivial Galois action. Since $\\op{G}'$ acts trivially\non $\\operatorname{Gal}(K(\\mu_{p^2} )\/K)$ it follows that $K(\\rho_2) \\cap K(\\mu_{p^2}) = K$. Thus conditions $\\eqref{71one}$ and $\\eqref{71two}$ are independent.\n\\par We show that the first three conditions may be simultaneously satisfied by considering the cases $K(\\rho_2)\\supseteq K_f$ and $K(\\rho_2)\\not \\supseteq K_f$ separately. First consider the case when $K(\\rho_2)\\supseteq K_f$. Let $r:=\\dim_{\\mathbb{F}_p} f(\\op{G}_K)$. Since $\\op{Gal}(K(\\rho_2)\/K)\\simeq \\operatorname{Ad}^0\\bar{\\rho}$, if $r< \\dim_{\\mathbb{F}_p} \\operatorname{Ad}^0\\bar{\\rho}$ the containment $K(\\rho_2)\\supset K_f$ is proper. Since $f$ is non-zero, Lemma $\\ref{l4}$ asserts that $K_f\\neq K$. Let $Q\\subset \\op{Gal}(K(\\rho_2)\/K)$ be the proper subgroup such that $\\op{Gal}(K(\\rho_2)\/K)\/Q\\simeq \\op{Gal}(K_f\/K)$. Lemma $\\ref{Pdecomposition}$ asserts that $Q$ decomposes into $\\mathbb{T}$-eigenspaces $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$ and Lemma $\\ref{fullrankLemma}$ asserts that the eigenspace $Q_{-2L_1}:=Q_{\\sigma_{-2L_1}}$ must be trivial. Hence the quotient $\\op{Gal}(K_f\/K)$ must have a non-zero $\\sigma_{-2L_1}$-eigenspace. Identify $\\op{Gal}(K_f\/K)$ with $f(\\op{G}_K)\\subset \\operatorname{Ad}^0\\bar{\\rho}$. Since $r<\\dim_{\\mathbb{F}_p} \\operatorname{Ad}^0\\bar{\\rho}$, Lemma $\\ref{fullrankLemma}$ asserts that $f(\\op{G}_K)_{-2L_1}=0$, a contradiction. Hence, $K(\\rho_2)\\supseteq K_f$ forces equality $K(\\rho_2)= K_f$. Let \\[\\alpha_1:=f_{\\restriction \\op{G}_K}:\\op{Gal}(K_f\/K)\\xrightarrow{\\sim} \\operatorname{Ad}^0\\bar{\\rho}\\]\nand \n\\[\\alpha_2:=\\rho_{2\\restriction \\op{G}_K}:\\op{Gal}(K_f\/K)\\xrightarrow{\\sim} \\operatorname{Ad}^0\\bar{\\rho}. \\] The composite $\\alpha_1\\alpha_2^{-1}$ is a $\\op{G}'$-automorphism of $\\operatorname{Ad}^0\\bar{\\rho}$. It follows from Corollary $\\ref{Coradd}$ that $\\alpha_1\\alpha_2^{-1}$ is a scalar $a\\in \\mathbb{F}_q^{\\times}$ and hence $\\alpha_1=a\\alpha_2$. Let $v$ satisfy $\\eqref{71one}$, $\\eqref{71two}$, $\\eqref{71four}$ and $\\eqref{71five}$ such that \n\\[(\\operatorname{Id}+X_{-2L_1})^{-1}\\rho_2(\\sigma_v)(\\operatorname{Id}+X_{-2L_1})\\in \\mathcal{T}\\]has non-trivial $H_1$ component. Since $v$ is a trivial prime, $\\sigma_v$ lies in $\\op{G}_K$. Identifying $\\op{ker}\\{\\operatorname{GSp}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\operatorname{GSp}(\\mathbb{F}_q)\\}$ with $\\operatorname{Ad}^0\\bar{\\rho}$, we view $\\rho_2(\\sigma_{v})$ as an element in $\\operatorname{Ad}^0\\bar{\\rho}$. Since $f(\\sigma_v)=a\\rho_2(\\sigma_v)$, we see that $(\\operatorname{Id}+X_{-2L_1})^{-1}f(\\sigma_v)(\\operatorname{Id}+X_{-2L_1})$ has non-zero $H_1$ component and hence is not contained in $\\mathfrak{t}_{2L_1}+\\op{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{2L_1})$. As a result, $f$ is not in $(\\operatorname{Id}+X_{-2L_1})\\mathcal{P}_v^{2L_1}(\\operatorname{Id}+X_{-2L_1})^{-1}$. It is easy to see that $f$ is not in $\\mathcal{N}_v$ and hence $\\eqref{71three}$ is also satisfied.\n\n\\par We consider the case when $K_f\\not\\subseteq K(\\rho_2)$. Since \\[[K(\\rho_2):K]=\\# \\operatorname{Ad}^0\\bar{\\rho} \\geq [K_f:K],\\]$K(\\rho_2)$ is not contained in $K_f$. It follows that $K(\\rho_2)\\supsetneq K(\\rho_2)\\cap K_f$ and $K_f\\supsetneq K(\\rho_2)\\cap K_f$ and thus by Lemma $\\ref{mainin}$, the images of \\[\\op{Gal}(K(\\rho_2)\/K(\\rho_2)\\cap K_f)\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\text{ and }\\op{Gal}(K_f\/K(\\rho_2)\\cap K_f)\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\] contain $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$. If $K_f\\subseteq K(\\rho_2,\\mu_{p^2})$, then it follows that\n\\[\\dim_{\\mathbb{F}_p} (\\op{Gal}(K(\\rho_2,\\mu_{p^2})\/K)_{\\sigma_{2L_1}})\\geq 2\\dim_{\\mathbb{F}_p} (\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}=2[\\mathbb{F}_q:\\mathbb{F}_p].\\]However, $\\op{Gal}(K(\\rho_2,\\mu_{p^2})\/K)_{\\sigma_{2L_1}}$ may be identified with \\[\\op{Gal}(K(\\rho_2)\/K)_{\\sigma_{2L_1}}\\simeq (\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}\\] since $K( \\mu_{p^2})$ contributes to the trivial eigenspace.\nHence, $K_f \\not \\subseteq K(\\rho_2, \\mu_{p^2} )$. Let $v$ be a prime satisfying conditions $\\eqref{71one}$, $\\eqref{71two}$, $\\eqref{71four}$ and $\\eqref{71five}$. Lemma $\\ref{mainin}$ asserts that the image of \\[\\op{Gal}(K_f\/K_f\\cap K(\\rho_2,\\mu_{p^2}))\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\] contains $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$ and thus, we have the freedom to stipulate that the $X_{2L_1}$-component of $f(\\sigma_v)$ be anything we like. Lemma $\\ref{lemma55}$ asserts that if $f\\in \\mathcal{N}_v$, an explicit relationship must be satisfied between the $X_{2L_1}$-component and the $H_1$-component of $f(\\sigma_v)$. It follows that we may alter the $X_{2L_1}$-component of $f(\\sigma_v)$ so that $f\\notin \\mathcal{N}_v$. Therefore all conditions may be satisfied and the proof is complete.\n\\end{proof}\n\\begin{Prop}\nThere is a finite set $Y\\supseteq Z$ such that $Y\\backslash S$ consists of trivial primes and $H^1_{\\mathcal{N^{\\perp}}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)=0$.\n\\end{Prop}\n\\begin{proof}\n\\par Let $Y$ be a finite set of primes containing $Z$ such that $Y\\backslash S$ consists of trivial primes. If $H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})\\neq 0$, we exhibit a trivial prime $v$ not contained in $Y$ such that \n\\[h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})h^1_{\\mathcal{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\n\\end{equation}\n\\par Consider the restriction maps\n\\begin{equation*}\n\\Phi_1:H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W})\\rightarrow \\bigoplus_{w\\in Y}H^1(\\op{G}_w, \\mathscr{W})\n\\end{equation*}\nand\n\\begin{equation*}\n\\Phi_2:H^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W})\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}).\n\\end{equation*} We show that the maps $\\Phi_1$ and $\\Phi_2$ have the same image. By the Poitou-Tate sequence,\n\\[0\\rightarrow \\op{image}(\\Phi_1)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W})\\rightarrow H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)^{\\vee},\\] i.e.,\nthe image of $\\Phi_1$ is the exact annihiliator of the image of the restriction map\n\\[H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}^*).\\]\nLet $\\mathfrak{M}$ be the Selmer condition \n\\[\\mathfrak{M}_w:=\\begin{cases}0\\text{ if }w\\in Y\\\\\nH^1(\\operatorname{G}_v, \\mathscr{W})\\text{ if }w=v\\\\\nH^1_{nr}(\\op{G}_w, \\mathscr{W})\\text{ if }w\\notin Y\\cup\\{v\\},\n\\end{cases}\\] with dual Selmer condition \n\\[\\mathfrak{M}_w^{\\perp}=\\begin{cases}H^1(\\operatorname{G}_v, \\mathscr{W}^*)\\text{ if }w\\in Y\\\\\n0\\text{ if }w=v\\\\\nH^1_{nr}(\\op{G}_w, \\mathscr{W}^*)\\text{ if }w\\notin Y\\cup\\{v\\}.\n\\end{cases}\\]\nBy the Poitou-Tate sequence, the image of $\\Phi_2$ is the exact annihilator of the restriction map \n\\[H^1_{\\mathfrak{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W}^*)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}^*).\\] By Proposition $\\ref{lastchebotarev}$ condition $\\eqref{71four}$, \n\\[H^1_{\\mathfrak{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W}^*)=H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)\\] and therefore, the image of $\\Phi_1$ is equal to the image of $\\Phi_2$.\n\\par We deduce that\n\\begin{equation}\\label{phi3phi4}\n\\begin{split}\n\\dim \\ker \\Phi_2-\\dim \\ker \\Phi_1 &=h^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\mathscr{W})-h^1(\\op{G}_{\\mathbb{Q},Y},\\mathscr{W})\\\\\n&=h^1(\\operatorname{G}_v, \\mathscr{W})-h^0(\\operatorname{G}_v, \\mathscr{W})\\\\\n&=h^1(\\operatorname{G}_v, \\mathscr{W})-h^1_{nr}(\\operatorname{G}_v, \\mathscr{W}).\\\\\n\\end{split}\n\\end{equation}\nBy $\\ref{phi3phi4}$, we deduce that the sequence\n\\begin{equation}\\label{lastequation}0\\rightarrow \\ker \\Phi_1\\rightarrow \\ker\\Phi_2\\rightarrow \\frac{H^1(\\operatorname{G}_v, \\mathscr{W})}{H^1_{nr}(\\operatorname{G}_v, \\mathscr{W})}\\rightarrow 0\\end{equation}\nis a short exact sequence.\n\n\\par Define the maps\n\\begin{equation*}\n\\Phi_3: H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}) \\rightarrow \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\n\\end{equation*}\nand \n\\begin{equation*}\n\\Phi_4: H^1(\\op{G}_{\\mathbb{Q}, Y},\\operatorname{Ad}^0\\bar{\\rho}) \\rightarrow \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}.\n\\end{equation*} From the Cassels-Poitou-Tate long exact sequence and the vanishing of $\\Sh^2_Y(\\operatorname{Ad}^0\\bar{\\rho})$, we deduce that the following sequences are exact\n\\begin{equation*}\n H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}) \\xrightarrow{\\Phi_3} \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\\rightarrow H^1_{\\mathcal{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}\\rightarrow 0\n\\end{equation*}\n\\begin{equation*}\n H^1(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}) \\xrightarrow{\\Phi_4} \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\\rightarrow H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}\\rightarrow 0.\n\\end{equation*}\nSet $t'$ to denote the difference $\\dim \\op{image} \\Phi_3-\\dim \\op{image} \\Phi_4$. From the assertion made in $\\eqref{dimgreaterone}$ we conclude that $t'\\geq 1$.\n\\par We claim that it suffices to find $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'+1$ elements in $\\ker\\Phi_3$, no linear combination of which lies in $\\mathcal{N}_v$. It follows then that the image of \\[\\ker\\Phi_3\\rightarrow \\frac{H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\] has dimension strictly greater than $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'$. From the exactness of \n\\[0\\rightarrow H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\ker\\Phi_3\\rightarrow \\frac{H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\]one may deduce that\n\\[\\begin{split}h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})<&\\dim \\ker \\Phi_3-\\dim \\operatorname{Ad}^0\\bar{\\rho}+t'.\\\\\n= & h^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})-\\dim \\operatorname{Ad}^0\\bar{\\rho} -\\dim \\op{im} \\Phi_4.\\\\\\end{split}\\]\nNote that $\\Sh^1_{Y}(\\operatorname{Ad}^0\\bar{\\rho})=0$ and thus an application of Wiles' formula \\eqref{wilesformula} shows that\n\\[\\begin{split}h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})=& h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\\n+&\\sum_{w\\in Y\\cup \\{\\infty\\}} (h^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}))\\end{split}\\]\nand \n\\[\\begin{split}h^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})=& h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\\n+&\\sum_{w\\in Y\\cup \\{v\\}\\cup \\{\\infty\\}} (h^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})).\\end{split}\\]\nTherefore, \n\\[\\begin{split}\n h^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})=& h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})+h^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\\\\n =&h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})+\\dim \\operatorname{Ad}^0\\bar{\\rho}.\n\\end{split}\\]\nTherefore, we have that \n\\[\\begin{split}h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})<& h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})-\\dim \\op{im} \\Phi_4\\\\\n=&\\dim \\op{ker} \\Phi_4\\\\\n=& h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}).\n\\end{split}\\]\nTherefore in order to complete the proof we proceed to construct $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'+1$ elements in $\\ker\\Phi_3$ no linear combination of which lies in $\\mathcal{N}_v$. We are in fact able to construct $\\dim \\operatorname{Ad}^0\\bar{\\rho}$ elements, which suffices since $t'\\geq 1$.\n\\par Note that $\\operatorname{Ad}^0\\bar{\\rho}\/\\mathscr{W}$ is isomorphic to $\\mathbb{F}_q(\\sigma_{-2L_1})$ and hence $H^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}\/\\mathscr{W})$ is zero. We find that $H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\mathscr{W})$ injects into $H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ and thereby it follows that $\\op{ker}\\Phi_2$ is contained in $\\op{ker}\\Phi_3$. Let $Z_1,\\dots, Z_s$ be a basis of $\\mathscr{W}$. By the exactness of $\\ref{lastequation}$ there exist $\\omega_i\\in \\text{ker}\\Phi_2$ such that $\\omega_i(\\tau_v)=Z_i$ for $i=1,\\dots, s$. We show that no linear combination of $\\{f, \\omega_1,\\dots, \\omega_s\\}$ lies in $\\mathcal{N}_v$. Let $Q=c_0 f+\\sum_{i=1}^s c_i \\omega_i$ be in $\\mathcal{N}_v$. Since $f$ is unramified at $v$, $f(\\tau_v)=0$. On the other hand,\n$Q(\\tau_v)=\\sum_{i=1}^s c_i Z_i$ is contained in $\\mathscr{W}$. Since $Q\\in \\mathcal{N}_v$, \\[Q(\\tau_v)=c'(\\operatorname{Id}+X_{-\\alpha})X_{\\alpha}(\\operatorname{Id}+X_{-\\alpha})^{-1}\\] for $\\alpha=2L_1$ and some constant $c'$. The root vectors $X_{\\alpha}$ and $X_{-\\alpha}$ are constant multiples of $e_{1,n+1}$ and $e_{n+1,1}$ respectively. Assume without loss of generality that $X_{\\alpha}=e_{1,n+1}$ and $X_{-\\alpha}=e_{n+1,1}$. Clearly, $X_{-\\alpha}^2=0$ and hence $(1+X_{-\\alpha})^{-1}=(1-X_{-\\alpha})$. We see that\n\\[\\begin{split}Q(\\tau_v)=&c'(\\operatorname{Id}+X_{-\\alpha})X_{\\alpha}(\\operatorname{Id}-X_{-\\alpha}) \\\\\n=& c'\\left(X_{\\alpha} +[X_{-\\alpha},X_{\\alpha}]-X_{-\\alpha}X_{\\alpha}X_{-\\alpha}\\right)\\\\\n=& c'\\left(e_{1,n+1}-H_1-e_{n+1,1}\\right). \\end{split}\\]We deduce that $Q(\\tau_v)=0$ since $e_{n+1,1}\\notin \\mathcal{W}$. Therefore, $c_i=0$ for all $i=1,\\dots, s$. As a consequence, $Q=c_0 f$. However, $f$ is not contained in $\\mathcal{N}_v$. It follows that $c_0=0$ and therefore, $Q=0$. Therefore no linear combination of $\\{f, \\omega_1,\\dots, \\omega_s\\}$ lies in $\\mathcal{N}_v$ and this completes the proof.\n\\end{proof}\nTo conclude the proof of Theorem $\\ref{main}$, we observe that on choosing an appropriately large choice of trivial primes the dual Selmer group vanishes and hence, by the lifting construction outlined in section $\\ref{section3}$, $\\rho_3$ lifts to a characteristic zero representation $\\rho$. Furthermore, $\\rho$ can be arranged to have similitude character $\\kappa$, and satisfy the local conditions $\\mathcal{C}_v$ at the primes $v\\in S$. Proposition $\\ref{bigimageprop}$ asserts that the image of $\\rho_2$ contains \\[\\widehat{\\op{Sp}}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2):=\\left\\{\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow\\op{Sp}_{2n}(\\mathbb{F}_q) \\right\\}\\]and it follows that $\\rho$ is irreducible.\n\\section{Examples}\\label{examples}\n\\par Let $p$ be an odd prime. Under certain hypotheses on $p$, we show that there are examples of reducible Galois representations \n$\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)$ which satisfy the conditions of Theorem $\\ref{main}$. First, we sketch the strategy used. The reader may refer to section $\\ref{notationsection}$ for some of the notation used in this section. Recall that $\\bar{\\chi}$ denotes the mod-$p$ cyclotomic character. Let $\\varphi_1,\\varphi_2$ and $\\bar{\\kappa}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GL}_1(\\mathbb{F}_q)$ be characters to be specificied later and $\\bar{r}$ the diagonal representation specified by \\[\\bar{r}:=\\left( {\\begin{array}{cccc}\n \\varphi_1 & & & \\\\\n & \\varphi_2 & & \\\\\n & & \\varphi_1^{-1} \\bar{\\kappa} & \\\\\n & & & \\varphi_2^{-1} \\bar{\\kappa} \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p).\\] The characters, $\\varphi_i$ and $\\bar{\\kappa}$ will be powers of $\\bar{\\chi}$.\n Let $D\\in \\op{GSp}_4(\\mathbb{Q}_p)$ be the diagonal matrix \\[D:=\\left( {\\begin{array}{cccc}\n p & & & \\\\\n & 1& & \\\\\n & & p^{-2} & \\\\\n & & & p^{-1}\n \\end{array} } \\right)\\] and set $H$ to denote $(D\\op{GSp}_4(\\mathbb{Z}_p)D^{-1})\\cap (D^{-1}\\op{GSp}_4(\\mathbb{Z}_p)D)$. Note that $H$ is the subgroup of $\\op{GSp}_4(\\mathbb{Z}_p)$ consisting of matrices \\[X=\\left( {\\begin{array}{cccc}\n a_{1,1} & p a_{1,2}& p^3 a_{1,3}& p^2 a_{1,4} \\\\\n p a_{2,1} & a_{2,2}& p^2 a_{2,3}& p a_{2,4}\\\\\n p^3 a_{3,1} & p^2 a_{3,2} & a_{3,3} & p a_{3,4}\\\\\n p^2 a_{4,1}& p a_{4,2} & p a_{4,3} & a_{4,4}\n \\end{array} } \\right).\\] Note that $D^{-1} X D$ is equal to \\[\\left( {\\begin{array}{cccc}\n a_{1,1} & a_{1,2}& a_{1,3}& a_{1,4} \\\\\n p^2 a_{2,1} & a_{2,2}& a_{2,3}& a_{2,4}\\\\\n p^6 a_{3,1} & p^4 a_{3,2} & a_{3,3} & p^2 a_{3,4}\\\\\n p^4 a_{4,1}& p^2 a_{4,2} & a_{4,3} & a_{4,4}\n \\end{array} } \\right)\\] and thus reduces to the Borel $\\op{B}(\\mathbb{F}_q)$ modulo $p$. Let $H_0$ denote the intersection of $H$ with $\\op{Sp}_4(\\mathbb{Z}_p)$. Recall that for $k\\geq 1$, $\\op{U}_k(\\mathbb{F}_p)\\subset \\op{B}(\\mathbb{F}_q)$ is the exponential subgroup generated by $\\operatorname{exp}((\\operatorname{Ad}^0\\bar{\\rho})_k)$. The strategy we adopt is as follows:\n \\begin{enumerate}\n \\item Under some conditions on $p$, we may choose $\\varphi_1,\\varphi_2$ and $\\bar{\\kappa}$ such that $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}},\\op{Ad}^0\\bar{r})$ is zero. Thus the global deformation problem (unramified outside $\\{p\\}$) is unobstructed.\n \\item We show that there is a lift\n \\[r:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{Z}_p)\\] of $\\bar{r}$ with image in $H$. Letting $\\bar{\\rho}$ denote the mod-$p$ reduction of $D^{-1} r D$, we note that the image of $\\bar{\\rho}$ is contained in $\\op{B}(\\mathbb{F}_p)$.\n \\item Let $\\Pi$ denote the intersection of the image of $\\bar{\\rho}$ with $\\op{U}_1(\\mathbb{F}_p)$. It is shown that after the mod-$p^2$ lift $r_2$ of $\\bar{r}$ may be carefully chosen so that $\\Pi$ surjects onto $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$. Lemma $\\ref{bigimagelemmaU1}$ shows that the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$. Moreover, the characters $\\varphi_1, \\varphi_2$ and $\\bar{\\kappa}$ are suitably chosen so that all the conditions of Theorem $\\ref{main}$ are satisfied.\n \\end{enumerate}\n Recall that $\\Phi^+$ consists of roots $\\{2L_1,2L_2, (L_1-L_2), (L_1+L_2)\\}$ and the simple roots are $\\lambda_1=L_1-L_2$ and $\\lambda_2=2L_2$. The root vectors are as follows \\[\n{\\small X_{2L_1}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & 1 & \\\\\n & 0 & & \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)}, {\\small X_{2L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & & \\\\\n & 0 & & 1 \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)},\\]\\[{\\small X_{L_1+L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & & 1 \\\\\n & 0 & 1 & \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)}\\text{ and }{\\small X_{L_1-L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & 1 & & \\\\\n & 0 & & \\\\\n & & 0 & \\\\\n & & -1 & 0\n \\end{array} }\\right).}\n\\] For $m\\geq 1$, set $H_0(\\mathbb{Z}\/p^m)$ (resp. $H(\\mathbb{Z}\/p^m)$) to denote the image of $H_0$ (resp. $H$) in $\\op{Sp}_4(\\mathbb{Z}\/p^m)$. Let $\\mathfrak{h}_m$ denote the kernel of the mod-$p^m$ reduction map $H_0(\\mathbb{Z}\/p^{m+1})\\rightarrow H_0(\\mathbb{Z}\/p^{m})$. Identify $\\mathfrak{h}_m$ with a subspace of $\\op{Ad}^0\\bar{r}$, so that $\\op{Id}+p^m X$ is identified with $X$ in $\\op{Ad}^0\\bar{r}$. It is easy to see that \n\\[\\mathfrak{h}_m=\\begin{cases}\n\\mathbb{F}_p\\langle H_1, H_2, X_{\\pm(L_1-L_2)}, X_{\\pm{2L_2}}\\rangle\\text{, for }m=1,\\\\\n\\mathbb{F}_p\\langle H_1, H_2, X_{\\pm(L_1-L_2)}, X_{\\pm{2L_2}}, X_{\\pm(L_1+L_2)}\\rangle\\text{, for }m=2,\\\\\n\\op{Ad}^0\\bar{r}\\text{, for }m\\geq 3.\n\n\\end{cases}\\]\n \\par Let $A$ be the Class group of $\\mathbb{Q}(\\mu_p)$ and let $\\mathcal{C}$ denote the mod-$p$ class group $\\mathcal{C}:=A\\otimes \\mathbb{F}_p$. The Galois group $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ acts on $\\mathcal{C}$ via the natural action. Since the order of $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows that $\\mathcal{C}$ decomposes into eigenspaces \n \\[\\mathcal{C}=\\bigoplus_{i=0}^{p-2} \\mathcal{C}(\\bar{\\chi}^i),\\]\n where $\\mathcal{C}(\\bar{\\chi}^i)=\\{x\\in \\mathcal{C} \\mid g\\cdot x=\\bar{\\chi}^{i}(g) x \\}$.\n \\begin{Lemma}\\label{lemma31}\n For $0\\leq i\\leq p-2$,\n \\begin{enumerate}\n \\item\\label{lemma31p1} the group $\\Sh^1_{\\{p\\}}(\\mathbb{F}_p(\\bar{\\chi}^i))$ injects into $\\op{Hom}(\\mathcal{C}(\\bar{\\chi}^i),\\mathbb{F}_p)$,\n \\item\\label{lemma31p2} the group $\\Sh^2_{\\{p\\}}(\\mathbb{F}_p(\\bar{\\chi}^i))$ equals zero if $\\mathcal{C}(\\bar{\\chi}^{p-i})$ equals zero.\n \\end{enumerate}\n \\end{Lemma}\n \\begin{proof}\n Since the order of $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows that \\[H^1(\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q}), \\mathbb{F}_p(\\bar{\\chi}^i))=0.\\] As a result, part $\\eqref{lemma31p1}$ follows from the inflation-restriction sequence. Part $\\eqref{lemma31p2}$ follows from part $\\eqref{lemma31p1}$ and Poitou-Tate duality for $\\Sh$-groups \\cite[Theorem 8.6.7]{NW}.\n \\end{proof}\n\n \\begin{Lemma}\\label{bigimagelemmaU1}\nSuppose that $p>2$ is a prime number and let $\\Pi$ be the a subgroup of $\\op{U}_1(\\mathbb{F}_p)$ such that the quotient map $\\Pi\\rightarrow \\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$ is surjective. Then $\\Pi$ is equal to $\\op{U}_1(\\mathbb{F}_p)$.\n \\end{Lemma}\n \\begin{proof}\n For $x,y\\in \\op{U}_1(\\mathbb{F}_p)$, set $\\{x,y\\}$ to denote the commutator $xyx^{-1}y^{-1}$. As $\\mathbb{F}_p$-vector spaces, we have that\n \\[\\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)=\\begin{cases}\n \\mathbb{F}_p\\langle \\op{exp}(X_{L_1-L_2}), \\op{exp}(X_{2L_2})\\rangle \\text{ if }k=1,\\\\\n \n \\mathbb{F}_p \\langle \\op{exp}(X_{L_1+L_2})\\rangle \\text{ if }k=2,\\\\\n \\mathbb{F}_p\\langle \\op{exp}(X_{2L_1}) \\rangle\\text{ if }k=3,\\\\\n 0\\text{ if }k>3.\n \n \\end{cases}\\]We check that if $x=\\op{exp}(X_{\\lambda})$ and $y=\\op{exp}(X_{\\mu})$, for roots $\\mu$ and $\\lambda$ in $\\Phi^+$, with height $k$ and $l$ respectively, then\n \\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\mod{\\op{U}_{k+l+1}(\\mathbb{F}_p)}.\\]We have the relations\n \\begin{equation}\\label{relationsrootvectors}[X_{L_1-L_2}, X_{2L_2}]=X_{L_1+L_2}\\text{ and } [X_{L_1-L_2}, X_{L_1+L_2}]=2X_{2L_1}\\end{equation} and that $X_{\\lambda}^2=0$ for $\\lambda\\in \\Phi^+$. We have therefore,\n \\[\\begin{split}\\{x,y\\}=&(\\op{Id}+X_{\\lambda})(\\op{Id}+X_{\\mu})(\\op{Id}-X_{\\lambda})(\\op{Id}-X_{\\mu})\\\\\n =&\\op{Id}+[X_{\\lambda}, X_{\\mu}]+(X_{\\mu}X_{\\lambda} X_{\\mu}-X_{\\lambda}X_{\\mu} X_{\\lambda})+(X_{\\lambda}X_{\\mu})^2.\\end{split}\\]Since $X_{\\lambda}^2$ and $X_{\\mu}^2$ are both equal to zero, we have that \n \\[X_{\\lambda} X_{\\mu} X_{\\lambda}=\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}]\\] and thus $X_{\\lambda} X_{\\mu} X_{\\lambda}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{2k+l}$. Likewise, the same reasoning shows that \\[X_{\\mu} X_{\\lambda} X_{\\mu}=\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}]\\] and therefore, $X_{\\mu} X_{\\lambda} X_{\\mu}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{k+2l}$. Next, observe that $(X_{\\lambda} X_{\\mu})^2$ is equal to $\\frac{1}{2}([X_{\\lambda}, X_{\\mu}])^2$. This too follows from the relations $X_{\\lambda}^2=X_{\\mu}^2=0$. From the relations $\\eqref{relationsrootvectors}$, we have that if $[X_{\\lambda}, X_{\\mu}]$ is nonzero, then, $\\lambda+\\mu$ is a root and there is a constant $c$ such that $[X_{\\lambda}, X_{\\mu}]=c X_{\\lambda+\\mu}$. Since, $X_{\\lambda+\\mu}^2=0$\n, it follows that $(X_{\\lambda} X_{\\mu})^2=0$. Since $X_{\\lambda} X_{\\mu} X_{\\lambda}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{2k+l}$, and the maximal height of any root is $3$, it follows that either $X_{\\lambda} X_{\\mu} X_{\\lambda}$ is zero, or a constant multiple of $X_{2L_1}$. It may be checked that $X_{2L_1} X_{\\lambda}=X_{\\lambda} X_{2L_1}=0$ for all $\\lambda\\in \\Phi^+$. As a consequence, we arrive at the following relation:\n\\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\op{exp}(-\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}])\\op{exp}(\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}]).\\] Note that $\\op{exp}(-\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}])$ and $\\op{exp}(\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}])$ are in $\\op{U}_{k+l+1}$. Therefore, we deduce that\n \\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\mod{\\op{U}_{k+l+1}(\\mathbb{F}_p)}.\\]\nWe deduce from the relations $\\eqref{relationsrootvectors}$ that the commutator $(x,y)\\mapsto\\{x,y\\}$ induces a surjective map:\n \\[\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)\\times \\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)\\rightarrow \\op{U}_{k+1}(\\mathbb{F}_p)\/\\op{U}_{k+2}(\\mathbb{F}_p).\\]\n It follows by ascending induction on $k$, that the quotient map \\[\\Pi\\cap \\op{U}_k(\\mathbb{F}_p)\\rightarrow \\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)\\] is surjective for $k\\geq 1$. By descending induction on $k$, we deduce that $\\Pi\\cap \\op{U}_k(\\mathbb{F}_p)=\\op{U}_k(\\mathbb{F}_p)$ for $k\\geq 1$. In particular, $\\Pi$ is equal to $\\op{U}_1(\\mathbb{F}_p)$ and the proof is complete.\n \\end{proof}\n \\begin{Prop}\n Let $p\\geq 23$ be a prime such that $\\mathcal{C}(\\bar{\\chi}^{p-i})=0$ for $i\\in \\{\\pm 3, \\pm 6, \\pm 9\\}$. There exists a Galois representation\n \\[\\bar{\\rho}:=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 &\\ast & \\ast & \\ast \\\\\n & 1 & \\ast & \\ast \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & \\ast & \\bar{\\chi}^9\n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{B}(\\mathbb{F}_p)\\] which satisfies the conditions of Theorem $\\ref{main}$. The similitude character of $\\bar{\\rho}$ is the odd character $\\bar{\\chi}^9$. Let $\\kappa$ be a fixed choice of a lift of $\\bar{\\kappa}$ such that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer divisible by $p(p-1)$ and $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$. There exists a finite set of auxiliary primes $X$ such that $p\\notin X$ and a lift $\\rho$ \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},\\{p\\}\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},\\{p\\}}$};\n \\node (GL2) [right of=GS]{$\\operatorname{GSp}_{4}(\\mathbb{F}_p).$};\n \\node (GL2W) [above of= GL2]{$\\operatorname{GSp}_{4}(\\mathbb{Z}_p)$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\rho$} (GL2W);\n \\end{tikzpicture}\\] for which \n\\begin{enumerate}\n\\item\\label{83p1} $\\rho$ is irreducible,\n\\item\\label{83p2} $\\rho$ is $p$-ordinary (in the sense of \\cite[section 4.1]{patrikisexceptional}),\n\\item\\label{83p3} $\\nu\\circ \\rho= \\kappa$.\n\\end{enumerate}\n \\end{Prop}\n \\begin{proof}\n We show a representation $\\bar{\\rho}$ satisfying the conditions of Theorem $\\ref{main}$ exists. It shall then follow from Theorem $\\ref{main}$ that there exists a lift $\\rho$ which satisfies the conditions $\\eqref{83p1}$, $\\eqref{83p2}$ and $\\eqref{83p3}$ above. Let $\\bar{r}$ be the representation with image in the diagonal torus:\n \\[\\bar{r}:=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 & & & \\\\\n & 1 & & \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & & \\bar{\\chi}^9 \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p).\\]\n \n The following matrix aids (in an informal way) in describing the eigenspace decomposition of $\\op{Ad}^0\\bar{r}$:\n \\[\\left( {\\begin{array}{cccc}\n 1 & \\bar{\\chi}^3 & \\bar{\\chi}^{-3} & \\bar{\\chi}^{-6}\\\\\n \\bar{\\chi}^{-3} & 1 & \\bar{\\chi}^{-6} & \\bar{\\chi}^{-9} \\\\\n \\bar{\\chi}^{3}& \\bar{\\chi}^{6}& 1 & \\bar{\\chi}^{-3} \\\\\n \\bar{\\chi}^{6}& \\bar{\\chi}^{9}& \\bar{\\chi}^3 & 1\n \\end{array} } \\right).\\]More precisely, $\\op{Ad}^0\\bar{r}$ is an $11$-dimensional space which decomposes into one-dimensional eigenspaces, and we have that\n \\[\\sigma_{\\pm 2L_1} = \\bar{\\chi}^{\\mp 3},\\sigma_{\\pm 2L_2} = \\bar{\\chi}^{\\mp 9},\\sigma_{\\pm(L_1+L_2)} = \\bar{\\chi}^{\\mp 6}\\text{ and }\\sigma_{\\pm (L_1-L_2)} = \\bar{\\chi}^{\\pm 3}. \\]\n We show that for $m\\geq 1$, the global cohomology group $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)$ is zero. As a Galois module, $\\mathfrak{h}_m$ decomposes into one-dimensional eigenspaces $\\mathbb{F}_p(\\sigma)$, where $\\sigma$ ranges through some of the characters $1,\\bar{\\chi}^{\\pm 3}, \\bar{\\chi}^{\\pm 6}, \\bar{\\chi}^{\\pm 9}$. It suffices to show that for the above choices of $\\sigma$, the cohomology group $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma))$ is zero. We show that $H^2(\\op{G}_p, \\mathbb{F}_p(\\sigma))$ is zero and $\\Sh^2_{\\{p\\}}(\\mathbb{F}_p(\\sigma))$ is zero. The dual $\\mathbb{F}_p(\\sigma)^*:=\\op{Hom}(\\mathbb{F}_p(\\sigma), \\mu_p)$ is isomorphic to $\\mathbb{F}_p(\\bar{\\chi}\\sigma^{-1})$. Since $p\\geq 13$, the character $\\bar{\\chi}\\sigma^{-1}_{\\restriction \\op{G}_p}\\neq 1$. It follows from local duality that $H^2(\\op{G}_p, \\mathbb{F}_p(\\sigma))$ is zero. It is a standard fact that $\\mathcal{C}(\\bar{\\chi})$ is zero, see \\cite[Proposition 6.16]{washington}. It follows from Lemma $\\ref{lemma31}$ and the assumptions on $p$ that $\\Sh^2_{\\{p\\}}(\\op{Ad}^0\\bar{r})$ is zero. We have thus shown that $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)$ is zero for all $m\\geq 1$.\n \\par We stipulate that all deformations of $\\bar{r}$ have similitude character equal to $\\chi^9$, where we recall that $\\chi$ denotes the cyclotomic character. For $m\\geq 1$, let $\\chi_m$ denote $\\chi$ modulo $p^m$. Recall that $\\mathfrak{h}_1$ is spanned by $H_1,H_2$ and $\\sigma_{\\pm \\lambda_i}$ for $i=1,2$. Since the characters $\\sigma_{\\lambda_i}$ for $i=1,2$ are both odd and \\[H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma_{\\lambda_i}))=0,\\] it follows from the global Euler characteristic formula \\cite[Theorem 8.7.4]{NW} that \\[\\op{dim}H^1(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma))=1.\\] Let $f_i$ be a generator for $H^1(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma_{\\lambda_i}))$ for $i=1,2$. Let $r_2'$ be the mod-$p^2$ lift \\[r_2':=\\left( {\\begin{array}{cccc}\n {\\chi}_2^3 & & & \\\\\n & 1 & & \\\\\n & & {\\chi}_2^6 & \\\\\n & & & {\\chi}_2^9 \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)\\] and $r_2$ be the twist $(\\op{Id}+p(f_1+f_2))r_2'$. Note that the image of $r_2$ is in $H(\\mathbb{Z}\/p^2)$. The obstruction to lifting $r_2$ to $r_3:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^3)$ is in $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_2)$, hence, is zero. Hence, $r_2$ lifts to $r_3$. Since $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)=0$ for all $m\\geq 1$, it follows that if $r_m:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^m)$ is a lift of $r_2$ (with similitude character $\\chi_m^9$), then $r_m$ lifts one more step to $r_{m+1}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^{m+1})$. Furthermore, the lift $r_{m+1}$ can be prescribed to have similitude character $\\chi_{m+1}^9$. The key ingredient here is that $H$ is a subgroup of $\\op{GSp}_4(\\mathbb{Z}_p)$. Since $H$ is a closed subgroup, it follows that $r_2$ lifts to a continuous characteristic zero representation $r:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}_p)$ with similitude character $\\chi^9$. Let $\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{B}(\\mathbb{F}_p)$ be the mod-$p$ reduction of $D^{-1} r D$ and let $\\Pi$ be $\\bar{\\rho}(\\op{G}_{\\mathbb{Q}(\\mu_p)})$. Lemma $\\ref{bigimagelemmaU1}$ asserts that if $\\Pi\\rightarrow \\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$ is surjective, then the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$. Let $\\Phi(r_2)\\subseteq \\mathfrak{h}_1$ be $r_2(\\op{ker} \\bar{r})$. In fact, $\\Phi(r_2)$ is contained in $\\mathbb{F}_p\\langle H_1, H_2, X_{L_1-L_2}, X_{2L_2}\\rangle $. Since the characters $1, \\sigma_{\\lambda_1}=\\bar{\\chi}^3, \\sigma_{\\lambda_2}=\\bar{\\chi}^{-9} $ are distinct, it follows that $\\Phi(r_2)$ decomposes into distinct eigenspaces\n \\[\\Phi(r_2)=\\Phi(r_2)^{\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})}\\oplus\\Phi(r_2)_{\\bar{\\chi}^3}\\oplus \\Phi(r_2)_{\\bar{\\chi}^{-9}} .\\]\n Since $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows from a straightforward application of the inflation restriction sequence that $f_{i\\restriction \\op{G}_{\\mathbb{Q}(\\mu_p)}}$ is nonzero. As a result, $\\Phi(r_2)_{\\bar{\\chi}^3}$ and $\\Phi(r_2)_{\\bar{\\chi}^{-9}}$ are nonzero, and hence, $X_{\\lambda_i}\\in \\Phi(r_2)$. Since $\\op{exp}(X_{\\lambda_1})$ and $\\op{exp}(X_{\\lambda_2})$ are generators of $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$, it follows that $\\Pi$ surjects onto $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$. Thus, the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$.\n \\par We show that the conditions of Theorem $\\ref{main}$ are satisfied.\n \\begin{itemize}\n \\item Condition $\\eqref{thc1}$ asserts that $p>4$, we have assumed that $p\\geq 23$.\n \\item Condition $\\eqref{thc2}$ asserts that $\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}$. Since $p>2$, up to conjugation, $\\bar{\\rho}(c)$ is equal to $\\left( {\\begin{array}{cccc}\n -1 & & & \\\\\n & 1 & & \\\\\n & & 1 & \\\\\n & & & -1\n \\end{array} } \\right)$. Explicit computation shows that w.r.t this basis, \n \\[(\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\mathbb{F}_p\\langle H_1, H_2, (L_1+L_2), -(L_2+L_2) \\rangle, \\]and hence, $\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}$ is equal to $4$. On the other hand, there are $4$ positive roots and the dimension of $\\mathfrak{n}$ is $4$.\n \\item Condition $\\eqref{thc3}$ asserts that the image of $\\bar{\\rho}$ contains the unipotent group $\\op{U}_1(\\mathbb{F}_p)$. This has been shown to be the case.\n \\item For condition $\\eqref{thc4}$, consider $\\sigma_{\\lambda}=\\bar{\\chi}^i$ and $\\sigma_{\\lambda'}=\\bar{\\chi}^j$. Since $i,j\\in\\{1, \\pm 3, \\pm 6, \\pm 9\\}$, we see that $|i-j|\\leq 182,$ this classification remains valid for homogeneous right coideal \nsubalgebras of the multiparameter version, see \\cite{Luz1, Tow}, of the Lusztig quantum group \n$u_q(\\frak{ sl}_{n+1}).$ We are reminded that any Hopf algebra generated by group-like and skew-\nprimitive elements is pointed, while in a pointed Hopf algebra the group-like elements span\nthe coradical, see \\cite[Definition 5.1.5]{Mon}.\n\nIn the second section we introduce main concepts and provide the general results \non the structure of the character Hopf algebras that are of use for classification. \nIn Lemma \\ref{qsim} we note that \nif the given character Hopf algebra $H=A\\# {\\bf k}[G]$ is a bosonisation\nof a quantum symmetric algebra $A,$ then each invariant \ndifferential subspace $U$ of $A$ defines a right coideal $U\\# {\\bf k}[G].$\nThis statement allows one to use noncommutative differential calculus, \n\\cite[p.6]{Luz2}, \\cite{MS}, \\cite{Kh1},\ndue to P. Schauenburg's characterization of quantum Borel subalgebras \\cite{Scha}.\nThe key point of the section is the construction of a PBW-basis\nover the coradical for a right coideal \nsubalgebra by means of \\cite{KhT, KhQ}. This basis, in particular, provides some \ninvariants for right coideal subalgebras (Definition \\ref{root}). \n\nIn the third section we define the multiparameter quantification of a Kac-Moody algebra as a character \nHopf algebra. This approach \\cite{Kh4} combines and generalizes all known quantifications. We do \nnot put unnecessary restrictions on the characteristic and on the quantification parameters. This allows \none, for example, to define a new class of finite Frobenius algebras as the Lusztig quantum groups \nover a finite field. All their right coideal subalgebras are also Frobenius \\cite{Skr} (finite Frobenius \nalgebras, in turn, have a significant r\\^{o}le in the coding theory \\cite{GMO}). \nIn Proposition \\ref{rtri} we provide \na short proof of the so called ``triangular decomposition\" in a quite general form.\n\nIn the fourth section (Proposition \\ref{phi}) we show that each homogeneous right coideal subalgebra\nin the quantum Borel algebra $U_q^+(\\frak{ sl}_{n+1})$ has PBW-generators over {\\bf k}$\\, [G]$\nof the following form\n\\begin{equation} \\Psi ^{\\hbox{\\bf s}}(k,m)=\n \\hbox{\\Large [[}\\ldots \\hbox{\\Large [}u[1+s_r,m], u[1+s_{r-1},s_r]\\hbox{\\Large ]}, \\ldots \\ ,\nu[1+s_1,s_2]\\hbox{\\Large ]},\\, u[k,s_1]\\hbox{\\Large ]},\n\\label{cbr1int}\n\\end{equation}\nwhere the brackets are defined by the structure \nof a character Hopf algebra, $[u,v]=uv-\\chi ^u(g_v)vu;$ \n$u[i,j]=[\\ldots [x_i,x_{i+1}],\\ldots ,x_j];$ $k\\leq s_12,$ then this is the case for homogeneous\nright coideal subalgebras of $u_q^{\\pm }(\\frak{ sl}_{n+1}).$ (If $q$ is not a root of 1 then\nall right coideal subalgebras that contain $G$ are homogeneous, Corollary \\ref{odn1}). \n\nIn Section 6 we consider right coideal subalgebras\nin the quantum Borel algebra that do not contain the coradical. \nNote that for every submonoid \n$\\Omega \\subseteq G$ the set of all linear combinations {\\bf k}$\\, [\\Omega]$\nis a right coideal subalgebra. We show that if the intersection $\\Omega $ of a homogeneous \nright coideal subalgebra $U$ with $G$ is a subgroup, then\n$U=\\, ${\\bf U}$_{\\theta }^{1}\\, ${\\bf k }$[\\Omega ].$ Here {\\bf U}$_{\\theta }^{1}$\nis a subalgebra generated by $g^{-1}_aa$ when $a$ runs through the described above \ngenerators of {\\bf U}$_{\\theta }.$\n\nIn Section 7 we characterize ad$_r$-invariant right coideal subalgebras that have \ntrivial intersection with the coradical in terms of K\\'eb\\'e's construction \\cite{Keb, Keb1}. \n\nWe see that the construction of {\\bf U}$_{\\theta }$ is completely\nconstructive, although it is not straightforward. \nHence by means of computer calculations one may find \nall necessary invariants of the coideal subalgebras and relations between them.\nIn the eighth section we provide a tableaux of the coideal subalgebras and their main\ncharacteristics for $n=3$ that was found by means of computer calculations.\n\nIn Sections 9-11 we consider the whole of $U_q(\\frak{ sl}_{n+1}).$\nThe triangular decomposition, \n\\begin{equation}\nU_q(\\frak{ sl}_{n+1})= U_q^-(\\frak{ sl}_{n+1})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_q^+(\\frak{ sl}_{n+1}),\n\\label{trint}\n\\end{equation}\nprovides a hope that any (homogeneous) right coideal subalgebra that contains the coradical\nhas the triangular decomposition as well, and for any two right coideal subalgebras \n$U_{\\theta }\\subseteq U_q^+ (\\frak{ sl}_{n+1}),$ $U_{\\theta ^{\\prime }}\\subseteq U_q^-(\\frak{ sl}_{n+1})$ \nthe tensor product \n\\begin{equation}\n \\hbox{\\bf U}\\, =U_{\\theta ^{\\prime }}\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_{\\theta }\n\\label{truint}\n\\end{equation}\nis a right coideal subalgebra. In this hypothesis just one statement fails, \nthe tensor product indeed is a right coideal but not always a subalgebra.\n\n\nTo describe conditions when (\\ref{truint}) is a subalgebra\nwe display the element $\\Psi ^{\\hbox{\\bf s}}(k,m)$\nschematically as a sequence of black and white points labeled by the numbers\n$k-1,$ $k,$ $k+1, \\ldots $ $m-1,$ $m,$ where the first point is always white, and\nthe last one is always black, while an intermediate point labeled by $i$ is black if and only if \n$i\\in {\\bf S}:$ \n$$\n \\stackrel{k-1}{\\circ } \\ \\ \\stackrel{k}{\\circ } \\ \\ \\stackrel{k+1}{\\circ } \n\\ \\ \\stackrel{k+2}{\\bullet }\\ \\ \\ \\stackrel{k+3}{\\circ }\\ \\cdots\n\\ \\ \\stackrel{m-2}{\\bullet } \\ \\ \\stackrel{m-1}{\\circ }\\ \\ \\stackrel{m}{\\bullet }\n$$\nConsider two elements \n$\\Psi ^{T_k}(k,\\tilde{\\theta }_k)$ and $\\Psi ^{T_i^{\\prime }}(i,\\tilde{\\theta }_i^{\\prime }),$\nwhere $T_k,$ $T_i^{\\prime }$ are defined as above by $\\theta $ and $\\theta ^{\\prime},$\nrespectively. Let us display these elements graphically\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots \\ & \\stackrel{i-1}{\\bullet } \n\\ & \\stackrel{i}{\\bullet }\\ \\ & \\stackrel{i+1}{\\circ }\\ & \\cdots &\n\\ & \\stackrel{\\tilde{\\theta }_k}{\\bullet } \\ & \\ & \\stackrel{\\tilde{\\theta }_i^{\\prime }}{\\cdot } \\cr\n\\ \\ & \\ \\ & \\circ \n\\ & \\circ \\ \\ & \\bullet \\ & \\cdots &\n\\ & \\bullet \\ & \\cdots \\ & \\bullet\n\\end{matrix}\n\\label{gra1int}\n\\end{equation}\nIn Theorem \\ref{osn5} we prove that (\\ref{truint}) is a subalgebra if and only if for each pair\n$(k, i),$ $1\\leq k,i\\leq n$ one of the following two options is fulfilled:\n\na) Representation (\\ref{gra1int}) has no fragments of the form \n$$\n\\begin{matrix}\n\\stackrel{t}{\\circ } \\ & \\cdots & \\stackrel{l}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet \n\\end{matrix}\n$$\n\nb) Representation (\\ref{gra1int}) has the form \n$$\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots & \\circ & \\cdots & \\bullet & \\cdots & \\stackrel{m}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet & \\cdots & \\circ & \\cdots & \\bullet \n\\end{matrix}\n$$\nwhere no one of the intermediate columns has points of the same color.\n\nThe obtained criterion allows use of the computer in order to find the total number $C_n$ of right coideal subalgebras which contain the coradical: \n$$\nC_2=26; \\ C_3=252; \\ C_4=3,368; \\ C_5=58,810; \\ C_6=1,290,930; \\ C_7=34,604,844.\n$$\n\n\\noindent\n{\\bf Remark}. If a Hopf algebra $H$ has a Hopf algebra pairing \n$\\langle , \\rangle : M\\times H\\rightarrow {\\bf k}$ with a Hopf algebra $M,$\nthen $M$ acts on $H$ via $m\\rightharpoonup h=\\sum h^{(1)}\\langle m, h^{(2)}\\rangle .$\nCertainly, in this case each right coideal is $M$-invariant. Conversely, if the pairing is left faithful\n(that is, $\\langle M, h\\rangle =0$ implies $h=0$) then each $M$-invariant subspace is a right coideal.\nFor $H=U_q(\\frak{sl}_{n+1})$ (or for $H=u_q(\\frak{sl}_{n+1})$ if $q^t=1$) there exists a Hopf algebra pairing with $M=GL_q(n),$ see \\cite{RTF, CM, Tow}.\nHence, alternatively, our main result provides a classification of $GL_q(n)$-invariant subalgebras that contain the coradical.\n\n\\smallskip\n\nThe computer part of this work has been done by the second author, \nwhile the proofs are due to the first one.\n\n\n\\section{Preliminaries}\n\\noindent\n{\\bf PBW-generators}.\nLet $S$ be an algebra over a field {\\bf k} and $K$ its subalgebra\nwith a fixed basis $\\{ g_j\\, |\\, j\\in J\\} .$ A linearly ordered subset $W\\subseteq S$ is said to be\na {\\it set of PBW-generators of $S$ over $K$} if there exists \na function $h:W\\rightarrow {\\bf Z}^+\\cup {\\infty },$\ncalled the {\\it height function}, such that the set of all products\n\\begin{equation}\ng_jw_1^{n_1}w_2^{n_2}\\cdots w_k^{n_k}, \n\\label{pbge}\n\\end{equation}\nwhere $j\\in J,\\ \\ w_11.$\nIn this case all elements $[x_i,x_j]=x_ix_j-p_{ij}x_jx_i,$ $|i-j|>1$ are skew primitive.\nTherefore the ideal of $G\\langle X\\rangle $ generated by these elements is a Hopf ideal.\nWe denote by ${{\\mathfrak A}}_n$ the quotient character \nHopf algebra, \n$$\n{{\\mathfrak A}}_n=G\\langle X\\, ||\\, [x_i,x_j]=0,\\ j-i>1 \\rangle \\stackrel{df}{=} \nG\\langle X\\rangle \/ {\\rm Id}\\, \\langle [x_i,x_j],\\ j-i>1\\rangle .\n$$\n\n\\begin{definition} \\rm\nThe elements $u,v$ are said to be \n{\\it separated} if there exists an index $j,$ $1\\leq j\\leq n,$\nsuch that either $u\\in {\\bf k}\\langle x_i\\ |\\ ij\\rangle $ or vise versa \n$u\\in {\\bf k}\\langle x_i\\ |\\ i>j\\rangle ,$\n$v\\in {\\bf k}\\langle x_i\\ |\\ i\nm_1^{\\prime }x_{i_1}+m_2^{\\prime }x_{i_2}+\\ldots +m_k^{\\prime }x_{i_k}\n\\label{ord}\n\\end{equation}\nif the first from the left nonzero number in\n$(m_1-m_1^{\\prime}, m_2-m_2^{\\prime}, \\ldots , m_k-m_k^{\\prime})$\nis positive, where $x_{i_1}>x_{i_2}>\\ldots >x_{i_k}$ in $X.$\nWe associate a formal degree $D(u)=\\sum _{x\\in X}m_xx\\in \\Gamma ^+$\nto a word $u$ in $G\\cup X,$ where $\\{ m_x\\, | \\, x\\in X\\}$ is the constitution of $u$\n(in \\cite[\\S 2.1]{FG} the formal sum $D(u)$ is called the {\\it weight} of $u$).\nRespectively, if $f=\\sum \\alpha _i u_i\\in G\\langle X\\rangle ,$ $0\\neq \\alpha _i\\in {\\bf k}$\nthen \n\\begin{equation}\nD(f)={\\rm max}_i\\{ D(u_i)\\} .\n\\label{degr}\n\\end{equation}\n\nOn the set of all words in $X$ we fix the lexicographical order\nwith the priority from the left to the right,\nwhere a proper beginning of a word is considered to \nbe greater than the word itself.\n\nA non-empty word $u$\nis called a {\\it standard} word (or {\\it Lyndon} word, or \n{\\it Lyndon-Shirshov} word) if $vw>wv$\nfor each decomposition $u=vw$ with non-empty $v,w.$\nA {\\it nonassociative} word is a word where brackets \n$[, ]$ somehow arranged to show how multiplication applies.\nIf $[u]$ denotes a nonassociative word then by $u$ we denote \nan associative word obtained from $[u]$ by removing the brackets\n(of course, $[u]$ is not uniquely defined by $u$ in general, however Lemma \\ref{ind}\nsays that the value of $[u]$ in ${{\\mathfrak A}}_n$ is uniquely defined provided that $u=u(k,m)$).\nThe set of {\\it standard nonassociative} words is the biggest set $SL$\nthat contains all variables $x_i$\nand satisfies the following properties.\n\n1)\\ If $[u]=[[v][w]]\\in SL$\nthen $[v],[w]\\in SL,$\nand $v>w$\nare standard.\n\n2)\\ If $[u]=[\\, [[v_1][v_2]]\\, [w]\\, ]\\in SL$ then $v_2\\leq w.$\n\n\\noindent\nEvery standard word has\nonly one alignment of brackets such that the appeared\nnonassociative word is standard (Shirshov theorem \\cite{pSh1}).\nIn order to find this alignment one may use the following\nprocedure: The factors $v, w$\nof the nonassociative decomposition $[u]=[[v][w]]$\nare the standard words such that $u=vw$\nand $v$ has the minimal length (\\cite{pSh2}, see also \\cite{Lot}).\n\\begin{definition} \\rm A {\\it super-letter}\nis a polynomial that equals a nonassociative standard word\nwhere the brackets mean (\\ref{sqo}).\nA {\\it super-word} is a word in super-letters. \nA $G$-{\\it super-word} is a super-word multiplied from the left by a group-like element. \n\\label{sup1}\n\\end{definition}\n\nBy Shirshov's theorem every standard word $u$\ndefines only one super-letter, in what follows we shall denote it by $[u].$\nThe order on the super-letters is defined in the natural way:\n$[u]>[v]\\iff u>v.$\n\\begin{definition} \\rm\nA super-letter $[u]$\nis called {\\it hard in }$H$\nprovided that its value in $H$\nis not a linear combination\nof values of super-words of the same degree (\\ref{degr})\nin smaller than $[u]$ super-letters, \n\\underline{and $G$-super-words of smaller degrees}.\n\\label{tv1}\n\\end{definition}\n\\begin{definition} \\rm\nWe say that a {\\it height} of a hard in $H$ super-letter $[u]$\nequals $h=h([u])$ if $h$\nis the smallest number such that: first, $p_{uu}$\nis a primitive $t$-th root of 1 and either $h=t$\nor $h=tl^r,$ where $l=$char({\\bf k}); and then the value in $H$\nof $[u]^h$\nis a linear combination of super-words of the same degree (\\ref{degr})\nin less than $[u]$ super-letters,\n\\underline{and $G$-super-words of smaller degrees}.\nIf there exists no such number then the height equals infinity.\n\\label{h1}\n\\end{definition}\nCertainly, if the algebra $H$ is homogeneous in each $a_i$ then one may omit\nthe underlined parts of the definitions.\n\\begin{theorem} $(${\\rm \\cite[{Theorem 2}]{Kh3}}$).$\nThe values of all hard in $H$ super-letters with the above defined height function\nform a set of PBW-generators for $H$ over {\\bf k}$[G].$\n\\label{BW}\n\\end{theorem}\n \n\\noindent \n{\\bf PBW-basis of a coideal subalgebra}. According to \\cite[Theorem 1.1]{KhT, KhQ}\nevery right coideal subalgebra {\\bf U} that contains all group-like elements has a PBW-basis\nover {\\bf k}$[G]$ which can be extended up to a PBW-basis of $H.$\n\nThe PBW-generators $T$ for {\\bf U} \ncan be obtained from the PBW-basis of $H$ given in Theorem \\ref{BW}\nin the following way. \n\nSuppose that for a given hard super-letter $[u]$ there exists an element $c\\in ${\\bf U}\nwith the leading term $[u]^s$ in the PBW-decomposition given in Theorem \\ref{BW}: \n\\begin{equation}\nc=[u]^s+\\sum \\alpha _iW_i+\\ldots \\in \\hbox{\\bf U},\n\\label{vad1}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $[u]$ super-letters, \n$D(W_i)=sD(u),$ and by the dots we denote a linear combination of $G$-super-words \nof $D$-degree less than $sD(u).$ We fix one of the elements with the minimal $s,$ and denote it by $c_u.$ Thus, for every hard in $H$ super-letter $[u]$ we have at most one element $c_u.$\nWe define the height function by means of the following lemma.\n\n\\begin{lemma}$\\!\\!\\! ($\\cite[{\\rm Lemma 4.3}]{KhT, KhQ}$).$ \nIn the representation $(\\ref{vad1})$ of the chosen element $c_u$\neither $s=1,$ or $p(u,u)$ is a primitive $t$-th root of $1$ and $s=t$ or \n$($in the case of positive characteristic$)$ $s=t({\\rm char}\\, {\\bf k})^r.$\n\\label{nco1}\n\\end{lemma}\nIf the height of $[u]$ in $H$ is infinite, then the height of $c_u$ in {\\bf U}\nis defined to be infinite as well. If the height of $[u]$ in $H$ equals $t,$ then, due to \nthe above lemma, $s=1$ (in the PBW-decomposition (\\ref{vad1}) the exponent \n$s$ must be less than\nthe height of $[u]$). In this case the height of $c_u$ in {\\bf U} is supposed to be $t$ as well.\nIf the characteristic $l$ is positive, and the height of $[u]$ in $H$ equals\n$tl^r,$ then we define the height of $c_u$ in {\\bf U} to be equal to $tl^r\/s$\n(thus, in characteristic zero the height of $c_u$ in {\\bf U} always\nequals the height of $[u]$ in $H$).\n\n\\begin{proposition}\nThe set of all chosen $c_u$ with the above defined height function\nforms a set of PBW-generators for {\\bf U} over {\\bf k}$[G].$ \n\\label{pro}\n\\end{proposition}\n\\begin{proof} See, \\cite[Proposition 4.4]{KhT, KhQ}. \\end{proof}\n\nWe note that there is an essential freedom in construction of the PBW-generators \nfor a right coideal subalgebra. In particular the PBW-basis is not uniquely defined in the above process. Nevertheless the set of leading terms of the PBW-generators indeed is uniquely defined.\n\n\\begin{definition} \\rm A hard super-letter $[u]$ is called {\\bf U}-{\\it effective} if there exists\n$c\\in \\, ${\\bf U} of the form (\\ref{vad1}). The degree $sD(u)\\in \\Gamma ^+ $ of $c$ with minimal $s$\nis said to be an {\\bf U}-{\\it root}. An {\\bf U}-root $\\gamma \\in \\Gamma ^+ $\n is called a {\\it simple} {\\bf U}-{\\it root} if it is not a sum of two or more other {\\bf U}-roots.\n\\label{root}\n\\end{definition}\nThus, the set of {\\bf U}-effective super-letters, the set of {\\bf U}-roots, and the set of \nsimple {\\bf U}-roots are invariants of any right coideal subalgebra {\\bf U}.\n\n{\\bf Remark}. There is already a fundamental for Lie theory notion of roots associated\nto semisimple Lie algebras. Certainly, the set of PBW-generators for \nthe universal enveloping algebra $U({\\frak g})$ \ncoincides with a basis of the Lie algebra ${\\frak g}.$ If we apply our definition to $U({\\frak g})$ then $U({\\frak g})$-roots are the formal degrees of basis elements related to a fixed set of generators\n$x_i, i\\in I.$ At the same time the formal degrees of basis elements for the Borel subalgebra \nare in one-to-one correspondence with positive roots: to each root \n$\\alpha _{i_1}+\\alpha _{i_2}+\\cdots +\\alpha _{i_k}$\ncorresponds a basis element $[\\ldots [x_{i_1},x_{i_2}],\\ldots ,x_{i_k}],$ \nsee \\cite[Chapter IV, \\S 3, Statement XVII]{Jac}. Therefore our definition of a root\nis a natural generalization of the classical notion. Probably the analogy would be \nmore clear if in our definition of the formal degree \nwe will replace the symbols $x_i$ with the characters $\\chi ^i$\nand identify the generators $g_i$ of the group $G$ with (exponents of the) basis elements $h_i$\nof the Cartan subalgebra since the classical roots are elements of the dual space\n$(\\sum_i {\\bf k}h_i)^*.$ Lemma \\ref{odn} below shows that this replacement is admissible.\nWe belive that by this very reason in \\cite{FG} the formal degree is referred to as {\\it weight},\nthe notion already well defined in the Lie theory. \n\n\\smallskip\n\\noindent \n{\\bf Differential calculi}. \nThe free algebra ${\\bf k}\\langle X\\rangle $ has a coordinate differential calculus\n\\begin{equation}\n\\partial_i(x_j)=\\delta _i^j,\\ \\ \\partial _i (uv)=\\partial _i(u)\\cdot v+\\chi ^u(g_i)u\\cdot \\partial _i(v).\n\\label{defdif}\n\\end{equation}\nThe partial derivatives connect the calculus with the coproduct on $G\\langle X\\rangle$ via\n\\begin{equation}\n\\Delta (u)\\equiv u\\otimes 1+\\sum _ig_i\\partial_i(u)\\otimes x_i\\ \\ \\ \n(\\hbox{mod }G\\langle X\\rangle \\otimes \\hbox{\\bf k}\\langle X\\rangle ^{(2)}),\n\\label{calc}\n\\end{equation}\nwhere ${\\bf k}\\langle X\\rangle ^{(2)}$ is the set (an ideal)\nof noncommutative polynomials without free and linear terms.\nSymetrically the equation\n\\begin{equation}\n\\Delta (u)\\equiv g_u\\otimes u+\\sum _ig_ug_i^{-1}x_i\\otimes\\partial _i^*(u)\\ \\ \\ \n(\\hbox{mod }G\\langle X\\rangle ^{(2)}\\otimes \\hbox{\\bf k}\\langle X\\rangle )\n\\label{calcdu}\n\\end{equation}\ndefines a dual differential calculus on ${\\bf k}\\langle X\\rangle $\nwhere the partial derivatives satisfy\n\\begin{equation}\n\\partial _j^*( x_i)=\\delta _i^j,\\ \\ \\partial _i^*(uv)=\n\\chi ^{i}(g_v) \\partial _i^*(u)\\cdot v+u\\cdot \\partial _i^*(v).\n\\label{difdu}\n\\end{equation}\nHere $G\\langle X\\rangle ^{(2)}{\\bf k}$ is an ideal of $G\\langle X\\rangle $\nof elements without free and linear terms.\nIf the kernel of $\\xi $ defined in (\\ref{gom}) is contained in $G\\langle X\\rangle ^{(2)}$ then formule\n(\\ref{calc}), (\\ref{calcdu}) with the $a$'s in place of the $x$'s define coordinate differential calculi\non the subalgebra $A$ of $H$ generated by the $a$'s. In this case restriction of $\\xi $\non $\\hbox{\\bf k}\\langle X\\rangle$ is a differential homomorphism, while\n (\\ref{calc}) and (\\ref{calcdu}) imply that each skew-primitive element $u$\nfrom $A^{(2)}=\\xi ({\\bf k}\\langle X\\rangle ^{(2)})$ is a constant with respect to both calculi,\n$\\partial _i(u)=\\partial _i^*(u)=0,$ $1\\leq i\\leq n.$\nMore details one can find in \\cite{MS, Kh1, Kh5}.\n\n\\smallskip\n\\noindent \n{\\bf Shuffle representation}. Let Ker$\\, \\xi \\subseteq G\\langle X\\rangle ^{(2)}.$ \nIn this case there exists a Hopf algebra projection $\\pi : H\\rightarrow {\\bf k}[G],$\n$a_i\\rightarrow 0, $ $g_i\\rightarrow g_i .$ Hence by the Radford theorem\n\\cite{Rad} we have a decomposition in a biproduct,\n$H=A\\# {\\bf k}[G],$ by means of the isomorphism\n$u\\rightarrow \\vartheta (u^{(1)})\\# \\pi(u^{(2)})$ with $\\vartheta (u)=\\sum _{(u)}u^{(1)}\\pi (S(u^{(2)})),$\nsee details in \\cite[\\S 1.5, \\S 1.7]{AS}.\n\nIf Ker$\\, \\xi $ is the biggest Hopf ideal\nin $G\\langle X\\rangle ^{(2)},$ or, equivalently, if $H$ is a Hopf algebra \nof type one in the sense of Nichols \\cite{Nic}, or, equivalently, \nif $A$ is a {\\it quantum symmetric algebra} (a Nichols algebra \\cite[\\S 1.3, Section 2]{AS}),\nthen $A$ has a shuffle representation as follows.\n\nThe algebra $A$ has a structure of a {\\it braided Hopf algebra}, \\cite{Tak1}, \nwith a braiding $\\tau (u\\otimes v)=p(v,u)^{-1}v\\otimes u.$\nThe braided coproduct \n$\\Delta ^b$ is connected with the coproduct on $H$ in the following way,\nsee \\cite[p. 93, (3.18)]{Kh5},\n\\begin{equation}\n\\Delta ^b(u)=\\sum _{(u)}u^{(1)}\\hbox{gr}(u^{(2)})^{-1}\\underline{\\otimes} u^{(2)}.\n\\label{copro}\n\\end{equation}\nAt the same time the tensor space $T(V),$ $V=\\sum_i {\\bf k}x_i$ also has a structure of a braided Hopf algebra.\nThis is the {\\it quantum shuffle algebra} $Sh_{\\tau }(V)$ with the coproduct \n\\begin{equation}\n\\Delta ^b(u)=\\sum _{i=0}^m(z_1\\ldots z_i)\\underline{\\otimes} (z_{i+1}\\ldots z_m),\n\\label{bcopro}\n\\end{equation}\nwhere $z_i\\in X,$ and $u=(z_1z_2\\ldots z_{m-1}z_m)$ is the tensor\n$z_1\\otimes z_2\\otimes \\ldots \\otimes z_{m-1}\\otimes z_m$ \nconsidered as an element of $Sh_{\\tau }(V).$\nThe map $a_i\\rightarrow (x_i)$ defines an embedding of the braided Hopf algebra\n$A$ into the the braided Hopf algebra $Sh_{\\tau }(V).$\nMore details can be find in \\cite{Nic, Wor, Scha, Ros, AG, FG1,Tak1, Flo, Kh5, Kh1}.\n\n\\smallskip\n\\noindent \n{\\bf Differential subalgebras}. \nIf {\\bf U} is a right coideal subalgebra of $H$ and {\\bf k}$[G]\\subseteq \\,${\\bf U}, then \n$\\vartheta (\\hbox{\\bf U})\\subseteq \\, ${\\bf U}, hence \n{\\bf U}$=U_A\\# {\\bf k}[G]$\nwith $U_A=\\vartheta (\\hbox{\\bf U})=\\hbox{\\bf U}\\cap A.$ Formula\n(\\ref{calc}) implies that $U_A$ is a differential subalgebra of $A,$\nthat certainly satisfies $gU_Ag^{-1}\\subseteq U_A,$ $g\\in G.$\nThe converse statement is valid if Ker$\\, \\xi $ is the biggest Hopf ideal.\n\\begin{lemma} Suppose that Ker$\\, \\xi $ is the biggest Hopf ideal\n in $G\\langle X\\rangle ^{(2)}.$\nIf $U$ is a differential subspace of $A={\\bf k}\\langle a_i\\rangle $ $=\\vartheta (H),$\nand $gUg^{-1}\\subseteq U,$ $g\\in G,$\nthen $U\\# {\\bf k}[G]$ is a right coideal of $H.$\n\\label{qsim}\n\\end{lemma}\n\\begin{proof}\nThe braided coproduct (\\ref{copro}) also defines a differential calculus \n\\begin{equation}\n\\Delta ^b(u)\\equiv u\\underline{\\otimes} 1+\n\\sum _i\\frac{\\partial ^bu}{\\partial x_i}\\underline{\\otimes} x_i\\ \\ \\ \n(\\hbox{mod }A \\underline{\\otimes} A^{(2)}).\n\\label{calc1}\n\\end{equation}\nIn \\cite[Theorem 4.8]{Kh1} this calculus is denoted by $d^*.$\nFormulae (\\ref{calc}), (\\ref{copro}), and (\\ref{calc1}) imply\n$\\partial ^bu\/\\partial x_i=g_i\\partial _i(u)g_i^{-1}.$\n\nSince $A$ has a representation as a subalgebra of the \nquantum shuffle algebra $Sh_{\\tau }(V),$ \nby \\cite[Theorem 5.1]{Kh1} applied to the calculus $d^*$\nthe restriction $\\Omega =\\xi |_{\\hbox{\\bf k}\\langle X\\rangle}$ has the following\ndifferential form\n\\begin{equation}\n\\Omega (u)=\\sum _{i_1,i_2,\\cdots ,i_n}\\frac{(\\partial^b) ^nu}{\\partial x_{i_1}\\partial x_{i_2}\\cdots \n\\partial x_{i_n}}(x_{i_n}x_{i_{n-1}}\\cdots x_{i_1}),\\ \\ \\ u\\in V^{\\otimes n},\n\\label{oderc3}\n\\end{equation} \nwhere as above $(x_{i_n}x_{i_{n-1}}\\cdots x_{i_1})$ is the tensor\n$x_{i_n}{\\otimes } x_{i_{n-1}}{\\otimes } \\cdots {\\otimes } x_{i_1}$\nconsidered as an element of $Sh_{\\tau }(V).$ By means of (\\ref{bcopro}) we have \n$$\n\\Delta ^b(\\Omega (u))=\\sum _{i_1,i_2,\\cdots ,i_n}\n\\frac{(\\partial ^b)^nu}{\\partial x_{i_1}\\partial x_{i_2}\\cdots \n\\partial x_{i_n}} \\sum _{k=1}^{n+1}(x_{i_n}\\cdots x_{i_k})\n\\underline{\\otimes }(x_{i_{k-1}}\\cdots x_{i_1})\n$$\n\n$$\n=\\sum _{k=1}^{n+1}\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\\left(\\sum _{i_k,i_{k+1},\\cdots ,i_n}\n\\frac{(\\partial ^b)^{n-k+1}\\left[ \\frac{(\\partial ^b)^{k-1}u}{\\partial x_{i_1} \\cdots \n\\partial x_{i_{k-1}}}\\right]}{\\partial x_{i_k} \\cdots \n\\partial x_{i_n}} (x_{i_n}\\cdots x_{i_k})\\right)\n\\underline {\\otimes }\n (x_{i_{k-1}}\\cdots x_{i_1})\n$$\n\\begin{equation}\n=\\sum _{k=1}^{n+1}\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\n\\Omega \\left( \\frac{(\\partial ^b)^{k-1} u}{\\partial x_{i_1} \\cdots \n\\partial x_{i_{k-1}}}\\right) \n\\underline {\\otimes }\n(x_{i_{k-1}}\\cdots x_{i_1}).\n\\label{oderc1}\n\\end{equation}\nSince $\\Omega $ is a $d^*$-differential map, we have got \n\\begin{equation}\n\\Delta ^b(w)=\\sum _{k=1}^{n+1}\n\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\n\\frac{(\\partial ^b)^{k-1} w}{\\partial x_{i_1}\\cdots \\partial x_{i_k}}\n\\underline {\\otimes }\n(x_{i_{k-1}}\\cdots x_{i_1}), \\ \\ w=\\Omega (u).\n\\label{oderc4}\n\\end{equation} \nThis formula implies that each differential subspace $W\\subseteq A$ with respect to $d^*$\n is a right coideal with respect to $\\Delta ^b.$\nIndeed, $\\Delta ^b(W)\\subseteq (A\\underline{\\otimes } A)\\cap \n(W\\underline{\\otimes } Sh_{\\tau})$ $=W\\underline {\\otimes } A.$\nSince\n$\\partial ^bu\/\\partial x_i=g_i\\partial_i(u)g_i^{-1},$\nthe space $U$ given in the lemma is a right coideal\nwith respect to the coproduct $\\Delta ^b.$ Now (\\ref{copro})\nshows that $U{\\bf k}[G]$ is a right coideal of $H.$ \nThe lemma is proved.\\end{proof}\n\n\\section{Multiparameter quantification of Kac-Moody algebras\\\\ as character Hopf algebras}\n\n\\noindent\n{\\bf Quantification of Borel subalgebras}.\nLet $C=||a_{ij}||$ be a symmetrizable by $D={\\rm diag }(d_1, \\ldots d_n)$ generalized Cartan matrix, $d_ia_{ij}=d_ja_{ji}.$\nDenote by $\\mathfrak g$ a Kac-Moody algebra defined by $C,$ see \\cite{Kac}.\nSuppose that the quantification parameters $p_{ij}=p(x_i,x_j)=\\chi ^i(g_j)$ are related by\n\\begin{equation}\np_{ii}=q^{d_i}, \\ \\ p_{ij}p_{ji}=q^{d_ia_{ij}},\\ \\ \\ 1\\leq i,j\\leq n. \n\\label{KM1}\n\\end{equation}\nIn this case the multiparameter quantization $U^+_q ({\\mathfrak g})$ of the \nBorel subalgebra ${\\mathfrak g}^+$\nis a character Hopf algebra defined by Serre relations \nwith the skew brackets in place of the Lie operation:\n\\begin{equation}\n[\\ldots [[x_i,\\underbrace{x_j],x_j], \\ldots ,x_j]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n.\n\\label{KM2}\n\\end{equation}\nBy \\cite[Theorem 6.1]{Khar} the left hand sides of these relations are skew-primitive elements \nin $G\\langle X\\rangle .$ Therefore the ideal generated by these elements is a Hopf ideal,\nwhile $U^+_q ({\\mathfrak g})$ indeed has a natural character Hopf algebra structure.\n\n\\begin{lemma} If $C$ is a Cartan matrix of finite type $($in particular the symmetric matrix $DC$ is positively defined $),$ and $q$ is not a root of $1$ then the grading of $U^+_q ({\\mathfrak g})$ by \ncharacters $(\\ref{grad})$ coincides with the grading by $\\Gamma ^+.$ \n\\label{odn}\n\\end{lemma}\n\\begin{proof} Since every homogeneous in $\\Gamma ^+$ element is homogeneous with respect to \n(\\ref{grad}), it suffices to show that the characters $\\chi ^i=\\chi ^{x_i},$ $1\\leq i\\leq n$\ngenerate a free Abelian group. Suppose in contrary that\n\\begin{equation}\n\\chi_1 \\stackrel{df}{=}(\\chi ^1)^{k_1}\\cdots (\\chi ^n)^{k_n}=(\\chi ^1)^{m_1}\\cdots (\\chi ^n)^{m_n}\n\\stackrel{df}{=} \\chi_2,\n\\label{fch}\n\\end{equation}\nwhere $k_i, m_i\\geq 0, \\ k_im_i=0,$ $1\\leq i\\leq n,$ and one of the $k_i$'s is nonzero.\nLet $g=g_1^{k_1}\\cdots g_n^{k_n},$ $h=g_1^{m_1}\\cdots g_n^{m_n}.$\nBy means of (\\ref{KM1}) we have\n$$\n\\chi _1(g)=\\prod _{1\\leq i,j\\leq n} p_{ij}^{k_jk_i}= \\prod _{i0.\n$$\nIn the same way $\\chi _2(h)=q^M,$ $M\\geq 0.$\nRelations (\\ref{KM1}) imply \n$$\n\\chi _2(g)\\chi _1(h)=\\prod _{1\\leq i,j\\leq n}p_{ij}^{m_ik_j}\\cdot \\prod _{1\\leq i,j\\leq n}p_{ij}^{m_jk_i}\n=\\prod _{1\\leq i, j\\leq n}(p_{ij}p_{ji})^{m_ik_j}=q^{L},\n$$\nwith $L=\\sum _{i,j}d_ia_{ij}m_ik_j \\leq 0$ since in the Catran matrix $a_{ij}\\leq 0$ for $i\\neq j,$\nwhile $k_im_i=0.$\n\nWe have $\\chi _1(g)=\\chi _2(g),$ and $\\chi_1(h)=\\chi _2(h).$ Therefore \n$q^{M+N}$ $=\\chi _1(g)\\chi _2(h)$ $=\\chi _2(g)\\chi _1(h)$ $=q^{L}.$ A contradiction.\n\\end{proof}\n\n\\smallskip\n\\noindent\n{\\bf Remark}. Of course, if the characters $\\chi_i,$ $1\\leq i\\leq n$ generate a \nfree Abelian group then $g_i,$ $1\\leq i\\leq n$ generate a free Abelian group as well.\nIn particular relations (\\ref{KM1}) imply that $G$ is a free Abelian group with the free generators\n$g_i,$ $1\\leq i\\leq n$ provided that $q$ is not a root of 1 and $C$ is of finite type.\n\\begin{corollary} \nIf $q$ is not a root of 1 and $C$ is of fifnite type, \nthen every subalgebra $U$ of $U_q^+({\\frak g})$ containing $G$\nis homogeneous with respect to each of the variables $x_i$.\n\\label{odn1}\n\\end{corollary}\n\\begin{proof} By the above lemma it suffices to note that $U$ is homogeneous \nwith respect to the grading by characters (\\ref{grad}). If $c=\\sum_ic_i\\in U$\nwith $c_i\\in H^{\\chi _i}$ and different $\\chi_i\\in G^*,$ then \n\\begin{equation}\ng^{-1}cg=\\sum _i \\chi_i(g)c_i\\in U, \\ \\ g\\in G.\n\\label{eqw}\n\\end{equation}\nAccording to the Dedekind Lemma there exist\nelements $h_i\\in G,$ such that the matrix $M=||\\chi_i(h_j)||$ is invertible.\nHence we may solve the system of equations (\\ref{eqw}) considering $c_i$\nas variables. In particular $c_i\\in U.$\n\\end{proof}\n\n\n\\smallskip\nIf the multiplicative order $t$ of $q$ is finite, \nthen we define $u^+_q ({\\mathfrak g})$ as $G\\langle X\\rangle\/{\\bf \\Lambda },$\nwhere ${\\bf \\Lambda }$ is the biggest Hopf ideal in $G\\langle X\\rangle ^{(2)}.$\nThis is a ${\\Gamma }^+$-homogeneous ideal, see \\cite[Lemma 2.2]{KA}. \nCertainly ${\\bf \\Lambda }$ contains all skew-primitive elements of\n$G\\langle X\\rangle ^{(2)}$ (each of them generates a Hopf ideal). Hence\nby \\cite[Theorem 6.1]{Khar} relations (\\ref{KM2}) are still valid in $u^+_q ({\\mathfrak g}).$\n\n\\smallskip\n\\noindent \n{\\bf Quantification of Kac-Moody algebras}.\nConsider a new set of variables $X^-=$ $\\{ x^-_1, x^-_2,\\ldots, x^-_n\\} .$ Suppose that an Abelian group $F$ generated by the elements $f_1, f_2, \\ldots ,f_n$ acts on the linear space spanned by \n$X^-$ so that $(x_i^-)^{f_j}=p_{ji}^{-1}x_i^-,$ where $p_{ij}$ are the same parameters, \nsee (\\ref{KM1}), that define $U_q^+(\\frak{g}).$ Relations (\\ref{KM1}) are invariant\nunder the substitutions $p_{ij}\\leftarrow p_{ji}^{-1},$ $q\\leftarrow q^{-1}.$ This allows us to define the\ncharacter Hopf algebra $U_q^-(\\frak{g})$ as $U_{q^{-1}}^+(\\frak{g})$ with the characters\n$\\chi ^i_-,$ $1\\leq i\\leq n$ such that $\\chi ^i_-(f_j)=p_{ji}^{-1}.$\n\nWe may extend the characters $\\chi ^i $ on $G\\times F$ in the following way\n\\begin{equation}\n\\chi ^i(f_j)\\stackrel{df}{=}p_{ji}=\\chi ^j(g_i).\n\\label{shar1}\n\\end{equation}\nIndeed, if $\\prod_k f_k^{m_k}=1$ in $F,$ then application to $x_i^-$\nimplies $\\prod_k p_{ki}^{-m_k}=1,$ hence $\\chi ^i(\\prod _k f_k^{m_k})=\\prod p_{ki}^{m_k}$\nequals 1 as well. In the same way we may extend the characters $\\chi ^i_-$ on $G\\times F$\nso that \n\\begin{equation}\n\\chi ^i_-=(\\chi ^i)^{-1} \\ \\ \\hbox{as characters of } G\\times F.\n\\label{shar2}\n\\end{equation}\n\n\nIn what follows we denote by $H$ a quotient group $(G\\times F)\/N,$ where \n$N$ is an arbitrary subgroup with $\\chi ^{i}(N)=1,$ $1\\leq i\\leq n.$ For example, if the quantification parameters satisfy additional symmetry conditions $p_{ij}=p_{ji},$ $1\\leq i,j\\leq n,$\nas this is a case for the original Drinfeld-Jimbo and Lusztig quantifications, then \n$\\chi ^i(g_k^{-1}f_k)=p_{ik}^{-1}p_{ki}=1,$ and we may take $N$ to be the subgroup generated by \n$g_k^{-1}f_k,$ $1\\leq k\\leq n. $ In this particular case the groups $H,$ $G,$ $F$ may be identified. \n\nIn the general case without loss of generality we may suppose that $G,F\\subseteq H.$\nCertainly $\\chi ^i, 1\\leq i\\leq n$ are characters of $H$ and $H$ still \nacts on the space spanned by $X\\cup X^-$ by means of these characters and their inverses. \n Consider the skew group algebra $H\\langle X\\cup X^-\\rangle $ as a character Hopf algebra:\n\\begin{equation}\n\\Delta (x_i)=x_i\\otimes 1+g_i\\otimes x_i,\\ \\ \\ \\Delta (x_i^-)=x_i^-\\otimes 1+f_i\\otimes x_i^-,\n\\label{AIcm}\n\\end{equation}\n\\begin{equation}\ng^{-1}x_ig=\\chi ^i(g)\\cdot x_i, \\ \\ g^{-1}x_i^-g=(\\chi ^i)^{-1}(g)\\cdot x_i^-, \\ \\ g\\in H.\n\\label{AIcm2}\n\\end{equation}\nWe define the algebra $U_q(\\frak{g})$ as a quotient\nof $H\\langle X\\cup X^-\\rangle $ by the following relations:\n\\begin{equation}\n[\\ldots [[x_i,\\underbrace{x_j],x_j], \\ldots ,x_j]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n;\n\\label{rela1}\n\\end{equation}\n\\begin{equation}\n[\\ldots [[x_i^-,\\underbrace{x_j^-],x_j^-], \\ldots ,x_j^-]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n;\n\\label{rela2}\n\\end{equation}\n\\begin{equation}\n[x_i, x_j^-]=\\delta_i^j(1-g_if_i), \\ \\ \\ \\ \\ 1\\leq i,j\\leq n\n\\label{rela3}\n\\end{equation}\nwhere the brackets are defined on $H\\langle X\\cup X^-\\rangle $ by the structure of character Hopf algebra, see (\\ref{sqo}). Since due to (\\ref{KM1}) and \\cite[Theorem 6.1]{Khar} all polynomials in the above relations are skew primitive in $H\\langle X\\cup X^-\\rangle ,$ they define a Hopf ideal of \n$H\\langle X\\cup X^-\\rangle $; that is, the natural homomorphism \n\\begin{equation}\nH\\langle X\\cup X^-\\rangle \\rightarrow U_q(\\frak{g})\n\\label{gom2}\n\\end{equation}\ndefines a Hopf algebra structure on $U_q(\\frak{g}).$\n\nIf $q$ has a finite multiplicative order then $u_q(\\frak{g})$ is defined by relations (\\ref{rela3})\nand $u=0,$ $u\\in {\\bf \\Lambda },$ $u^-=0,$ $u^-\\in {\\bf \\Lambda }^-,$ where ${\\bf \\Lambda },$\n${\\bf \\Lambda }^-$ are the biggest Hopf ideals respectively in $G\\langle X\\rangle ^{(2)}$\nand $F\\langle X^-\\rangle ^{(2)}.$\n\nBoth algebras $U_q(\\mathfrak{g}),$ and $u_q(\\mathfrak{g})$ have a grading by the additive \ngroup $\\Gamma $ generated by $\\Gamma ^+,$\nsee p.\\pageref{Gamma}, provided that we put $D(x_i^-)=-D(x_i)=-x_i,$ $D(H)=0$ \nsince in this way relations (\\ref{rela3}) became homogeneous. \n\n\\begin{corollary} \nIf $q$ is not a root of 1 and the Cartan matrix $C=||a_{ij}||$ is of finite type then every subalgebra \n$U$ of $U_q(\\frak{g})$ containing $H$ is $\\Gamma $-homogeneous.\n\\label{odn11}\n\\end{corollary}\n\\begin{proof}\nBy Lemma \\ref{odn} \nand definition (\\ref{shar2}) grading by $\\Gamma $ coincides with the grading\n(\\ref{grad}) by the group of characters (freely) generated by $\\chi ^i,$ $1\\leq i\\leq n.$\nHence every subspace invariant under the conjugations by $H$ is \n$\\Gamma $-homogeneous. \n\\end{proof}\n\n\\smallskip\nThe defined quantification reduces to known ones under a suitable choice of \n$x_i, x_i^-$ depending up the particular definition of $U_q (\\frak{g}).$ For example \nfor classical case of one parameter quantification we have $G=F=H,$ and in the \nnotations of \\cite{Luz2} we may identify\n$$\nx_i= E_i,\\ g_i= K_i,\\ x_i^-= \nF_iK_i(v^{-d_i}-v^{d_i})^{-1}, \\ p_{ij}= v^{-d_ia_{ij}}, \n$$\nwhile in the notations of \\cite{Luz1, Mul} we may take\n$$\nx_i= E_i,\\ g_i=\\tilde{K}_i, \\ x_i^-= F_i\\tilde{K}_i(v_i^{-1}-v_i)^{-1}, \\\np_{i\\mu }= v^{-\\langle \\mu ,i^{\\prime }\\rangle }.\n$$\nFor two-parameter quantizations, say in the notations of \\cite{BGH}, we may put\n$$\nx_i\\leftarrow e_i, \\ g_i\\leftarrow \\omega _i, \\ x_i^-\\leftarrow f_i(\\omega _i^{\\prime })^{-1}(r_i-s_i)^{-1},\\ \nf_i\\leftarrow (\\omega _i^{\\prime })^{-1},\n$$\nand find values of parameters $p_{ij}$ by means of \\cite[(B2), (C2), (D2)]{BGH}.\nFor the multiparameter case of Reshetikhin or DeConcini-Kac-Procesi \nin the notations of \\cite{CV}, we may take \n$$\nx_i\\leftarrow E_iL_{\\beta _i}, \\ g_i\\leftarrow L_{\\beta _i-\\alpha _i+\\gamma _i}, \\ x_i^-\\leftarrow F_iL_{\\alpha _i+\\beta _i}^{-1}(q_i-q_i^{-1})^{-1},\\ f_i\\leftarrow L_{\\gamma _i+\\alpha _i+\\beta _i}^{-1}.\n$$\n\n\\smallskip\n\\noindent\n{\\bf Triangular decomposition}.\nOne may prove that the subalgebra of \n$U_q (\\frak{g})$ generated by $G$ and values of $x_i,$ $1\\leq i\\leq n$ is isomorphic to\n $U_q^+ (\\frak{g})$ \nwhile the subalgebra generated by $F$ and values of $x_i^-,$ \n$1\\leq i\\leq n$ is isomorphic to $U_q^- (\\frak{g}).$ \nMoreover, one has the following so called ``triangular decomposition'' for both algebras:\n\\begin{equation}\nU_q(\\frak{g})= U_q^-(\\frak{g})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_q^+(\\frak{g}),\n\\label{tr}\n\\end{equation}\n\\begin{equation}\nu_q(\\frak{g})= u_q^-(\\frak{g})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}u_q^+(\\frak{g}).\n\\label{tr1}\n\\end{equation}\nActually this is not so evident (see \\cite{Luz1, Luz2} for standard one parameter version, \n\\cite{CV} for the multiparameter version with Cartan matrix of finite type,\n \\cite{BGH} for two-parameter version with particular Cartan matrices only). \nWe shall provide here a relatively short proof in the general setting that uses\na lemma on tensor decomposition\nfor character Hopf algebras, \\cite[Lemma 6.2]{Kh4}, and (in case $q^t=1$) the \nHeyneman--Radford theorem.\n\\begin{proposition}\nLet $J\\subseteq G\\langle X\\rangle ^{(2)} ,$ $J^-\\subseteq F\\langle X^-\\rangle ^{(2)}$\nbe constitution homogeneous Hopf ideals of \n$G\\langle X\\rangle $ and $F\\langle X^-\\rangle $ respectively.\nDenote by ${\\mathfrak A}$ the algebra generated over $H$ by $X\\cup X^-$ and defined by the relations $(\\ref{rela3})$ and\n$u_s=0,$ $s\\in S,$ $u^-_t=0,$ $t\\in T,$ where $\\{ u_s,$ $s\\in S\\} $ \n$($respectively $\\{ u_t^-,$ $t\\in T\\} )$ \nis a set of homogeneous generators of the ideal $J$ $($respectively $J^-).$ We have\n\\begin{equation}\n{\\mathfrak A}= (F\\langle X^-\\rangle\/J^-)\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}(G\\langle X\\rangle\/J).\n\\label{tria}\n\\end{equation}\n\\label{rtri}\n\\end{proposition}\n\\begin{proof}\nWe note first that the algebra ${\\mathfrak A}_1$ generated over $H$ by $X$ and defined by the relations\n$u_s=0,$ $s\\in S$ has the form ${\\bf k}[H]\\otimes _{{\\bf k}[G]}(G\\langle X\\rangle\/J),$\nwhile the algebra ${\\mathfrak A}_2$ generated over $H$ by $X^-$ and defined by the relations\n$u^-_t=0,$ $t\\in T$ has the form $(F\\langle X^-\\rangle\/J^-)\\otimes _{{\\bf k}[F]}{\\bf k}[H].$\nHence it suffices to show that ${\\mathfrak A}={\\mathfrak A}_2\\otimes _{{\\bf k}[H]}{\\mathfrak A}_1.$ \n\nDenote by $D_i,$ $D_i^- ,$ $1\\leq i\\leq n$ the linear maps \n$$D_i:{\\bf k}\\langle X^-\\rangle \\rightarrow H\\langle X^-\\rangle ,\\ \\ \\ \nD_i^-:{\\bf k}\\langle X\\rangle \\rightarrow H\\langle X\\rangle $$\nthat satisfy the initial conditions \n\\begin{equation}\nD_i(x_j^-)=D_j^-(x_i)=\\delta_i^j(1-g_if_i),\n\\label{inc}\n\\end{equation}\nand the skew differential Leibniz rules\n\\begin{equation}\nD_i(v^-\\cdot w^-)=D_i(v^-)\\cdot w^-+p(x_i, v^-)v\\cdot D_i(w^-),\\ \\ \\ v^-, w^-\\in {\\bf k}\\langle X^-\\rangle ;\n\\end{equation}\n\\begin{equation}\nD_i^-(u\\cdot v)=p(v,x_i^-)D_i^-(u)\\cdot v+u\\cdot D_i^-(v), \\ \\ \\ u, v\\in {\\bf k}\\langle X\\rangle .\n\\label{sqdf2}\n\\end{equation}\nLemma 6.2 \\cite{Kh4} (under the substitutions \n$k\\leftarrow n,$ $n\\leftarrow 2n,$ $x_{n+i}\\leftarrow x_i^-,$ $G\\leftarrow H,$ \n$H\\leftarrow {\\mathfrak A})$ \ngives the required decomposition provided that there exist homogeneous defining\nrelations $\\{ \\varphi _s=0,$ $s\\in S\\} ,$ and $\\{ \\psi _t=0,$ $t\\in T\\} $\n for ${\\mathfrak A}_1$ and ${\\mathfrak A}_2$ respectively, such that \n\\begin{equation}\nD_i(\\psi_t)\\in H\\cdot J^-,\\ \\ D_i^-(\\varphi _s)\\in H\\cdot J, \\ \\ \\ 1\\leq i\\leq n,\\ s\\in S,\\ t\\in T.\n\\label{inw}\n\\end{equation}\nConsider the linear maps\n\\begin{equation}\n\\tilde{D}_i^-: u\\rightarrow \\partial_ i^*(u)\n-p_{ii}^{-1}p(u,x_i)\\partial_i(u)g_if_i, \\ \\ \\ u\\in{\\bf k}\\langle X\\rangle ,\n\\label{sqi1}\n\\end{equation}\nwhere the partial derivatives are defined in (\\ref{calc}) and (\\ref{calcdu}). We have \n$\\tilde{D}_i^-(x_j)=\\delta _i^j(1-g_if_i),$ while relations (\\ref{defdif}) and (\\ref{difdu}) imply\nthe differential Leibniz rule \n$\\tilde{D}_i^-(u\\cdot v)=p(x_i,v)\\tilde{D}_i^-(u)\\cdot v+u\\cdot \\tilde{D}_i^-(v).$ Since \naccording to (\\ref{shar1}) we have $p(x_i,v)=p(v, x_i^-),$ the Leibniz rule and initial values for \n$D_i^-$ coincides with that for $\\tilde{D}_i^-.$ Hence $D_i^-=\\tilde{D}_i^-.$ In perfect analogy we have\n\\begin{equation}\nD_i(u^-)=\\partial_{-i}^*(u^-)p(x_i,u^-)p_{ii}^{-1}-g_if_i \\partial_{-i}(u^-),\n \\ \\ \\ u^-\\in{\\bf k}\\langle X^-\\rangle ,\n\\label{sqi2}\n\\end{equation}\nwhere $\\partial _{-i},$ $\\partial _{-i}^* $ are left and right partial derivatives on\n ${\\bf k}\\langle X^-\\rangle $ with respect to $x_i^-.$\n\nNow if $u_s,$ $s\\in S$ and $u_t^-,$ $t\\in T$ are skew primitive elements\n(as this is the case for ${\\mathfrak A}=U_q(\\frak{g})$) then they are constants \nfor $\\partial _i,$ $\\partial _i^*,$ and $\\partial _{-i},$ $\\partial _{-i}^*,$\nrespectively.\nHence (\\ref{sqi1}), (\\ref{sqi2}) imply $D_i^-(u_s)=D_i(u_t^-)=0$ and \\cite[Lemma 6.2]{Kh4} applies.\n\nIn the general case by the Heyneman-Radford theorem (see \\cite[Corollary 5.4.7]{Mon} or very ``skew \nprimitive\" version \\cite[Corollary 5.3]{Kh2}) the Hopf ideal $J$ has a nonzero\nskew primitive element provided that $J\\neq 0.$ Denote by $J_1$ an ideal generated\nby all skew primitive elements of $J.$ Clearly $J_1$ is a Hopf ideal. \nSince all homogeneous components of a skew primitive \nelement are skew primitive, the Hopf ideal $J_1$ is homogeneous. Moreover, we have\n$D_i^-(u_s)=0,$ $s\\in S_1$ where $u_s$ run through the set of all homogeneous skew primitive \nelements of $J.$ Now consider the Hopf ideal $J\/J_1$ of the quotient Hopf algebra\n$G\\langle X\\rangle \/J_1.$ If $J\\neq J_1$ then this ideal also has nonzero skew primitive elements.\nDenote by $J_2\/J_1$ the ideal generated by all skew primitive elements of $J\/J_1,$\nwhere $J_2$ is its preimage with respect to the natural homomorphism\n$G\\langle X\\rangle \\rightarrow G\\langle X\\rangle \/J_1.$ Again we have \n$D_i^-(\\bar{u}_s)=0,$ $s\\in S_2$ in $G\\langle X\\rangle \/J_1,$ where $\\bar{u}_s$ \nrun through the set of all homogeneous skew primitive elements of $J\/J_1.$\nIn particular the ideal $J_2$ has a set of generators $u_s,$ $s\\in S_1\\cup S_2$\nsuch that $D_i^-(u_s)\\in J_1.$ Continuing the process we shall find a set of generators \n$u_s,$ $s\\in S_1\\cup S_2\\cup S_3\\cup \\ldots $ for $J$ such that $D_i^-(u_s)\\in J,$ all $s.$\n\nIn perfect analogy we find a set of generators $u_t^-,$ \nfor $J^-$ such that $D_i(u_t^-)\\in J^-,$ all $t.$ Hence \\cite[Lemma 6.2]{Kh4} applies. \n\\end{proof}\n\n\\noindent\n{\\bf Remark}. In the proof we do not use relations (\\ref{KM1}) on the quantification parameters, while relations (\\ref{shar1}), (\\ref{shar2}) on the characters are essential. Also we are reminded that originally the maps $D_i,$ $D_i^-$ were defined so that $D_i(u^-)=[x_i,u^-],$ $D_i^-(u)=[u,x_i^-]$ in the algebra \n$H\\langle X\\cup X^-|| [x_i,x_j^-]=\\delta _i^j(1-g_if_i)\\rangle .$ Hence equalities $D_i^-=\\tilde{D}_i^-$\nand (\\ref{sqi2}) imply a differential representation:\n\\begin{equation}\n[x_i,u^-]=\\partial_{-i}^*(u^-)p(x_i,u^-)p_{ii}^{-1}-g_if_i \\partial_{-i}(u^-),\\ \\ \\ u^-\\in{\\bf k}\\langle X^-\\rangle ,\n\\label{sqi3}\n\\end{equation}\n\\begin{equation}\n[u, x_i^-]= \\partial_ i^*(u)-p_{ii}^{-1}p(u,x_i)\\partial_i(u)g_if_i, \\ \\ \\ u\\in{\\bf k}\\langle X\\rangle .\n\\label{sqi4}\n\\end{equation}\n\n\n\\section{PBW-generators for coideal subalgebras in \n$U_q^+(\\frak{sl}_{n+1}),$ $u_q^+(\\frak{sl}_{n+1})$}\nSuppose that the quantification parameters $p_{ij}=p(x_i,x_j)=\\chi ^{i}(g_{j})$\nare connected by the following relations \n\\begin{equation}\np_{ii}=q; \\ \\ p_{i\\, i+1}p_{i+1\\, i}=q^{-1}; \\ \\ \np_{ij}p_{ji}=1,\\ |i-j|>1,\n\\label{a1rel}\n\\end{equation}\nwhere $q\\not= \\pm 1.$ By definition \n $U_q^+(\\frak{ sl}_{n+1})$ as a character Hopf algebra is set up by the relations\n\\begin{equation}\n[[x_i,x_{i+1}],x_{i+1}]=[x_i,[x_i,x_{i+1}]]=[x_i,x_j]=0, \\ \\ |i-j|>1.\n\\label{rela}\n\\end{equation}\nThe structure of this algebra is defined by the following theorem (see, \\cite[Theorem $A_n$]{Kh4},\nor in other terms \\cite[Lemmas pp. 176, 184]{CM}, \\cite{CLMT}). \nRecall that $u(k,m)$ $=x_kx_{k+1}\\ldots x_m,$ $k\\leq m.$\nThe standard word $u(k,m)$ defines a super-letter\n$u[k,m]\\stackrel{df}{=}$ $[x_k[x_{k+1}[\\ldots[x_{m-1}x_m]\\ldots ]]],$\nwhile by definition $u[k,k]=x_k.$ Of course the value of $u[k,m]$ in\n$U_q^+(\\frak{ sl}_{n+1})$ is independent of the alignment of brackets (Lemma \\ref{ind}). \nBy $U_q^+(\\frak{ sl}_{n+1})^{(2)}$\nwe denote the ideal $\\xi (G\\langle X\\rangle ^{(2)})$ \ngenerated by the values of $x_ix_j,$ $1\\leq i,j\\leq n.$\n\\begin{theorem}\n$1.$ The values of the super-letters $u[k,m],$ $1\\leq k\\leq m\\leq n$ in $U_q^+(\\frak{ sl}_{n+1})$\nform the set of PBW-generators of $U_q^+(\\frak{ sl}_{n+1})$ over ${\\bf k}[G].$ All heights are infinite.\n\n$2.$ If $q$ is not a root of unity, then the ideal $U_q^+(\\frak{ sl}_{n+1})^{(2)}$ has no nonzero skew-primitive elements.\n\n$3.$ If $u$ is a standard word then either $u=u(k,m)$ or $[u]=0$ in $U_q^+(\\frak{ sl}_{n+1}).$\n\\label{strA}\n\\end{theorem}\nAccording to the Heyneman-Radford theorem (see \\cite{HR}, or \\cite[Corollary 5.4.7]{Mon}) every non-zero bi-ideal of a character Hopf algebra always has nonzero skew primitive elements. By this reason the second statement \nimplies that Ker$\\, \\xi ,$ $\\ \\xi :G\\langle X\\rangle \\rightarrow U_q^+(\\frak{ sl}_{n+1}),$ is the biggest\nHopf ideal in $G\\langle X\\rangle ^{(2)}.$ In particular one can apply Lemma \\ref{qsim}\nto $U_q^+(\\frak{ sl}_{n+1})$ provided $q$ is not a root of 1.\n\nIf $q$ is a root of 1 ($q\\neq \\pm 1$), then by definition\n$u_q^+(\\frak{ sl}_{n+1})$ is a quotient $U_q^+(\\frak{ sl}_{n+1})\/{ \\Lambda },$ where\n$\\Lambda $ is the biggest Hopf subideal of $U_q^+(\\frak{ sl}_{n+1})^{(2)}.$\nHence we may apply Lemma \\ref{qsim} to $u_q^+(\\frak{ sl}_{n+1})$ as well.\n\n\\smallskip\nIf {\\bf U} is a right coideal subalgebra of $U_q^+(\\frak{ sl}_{n+1})$\nthat contains {\\bf k}$\\, [G],$ then by Proposition \\ref{pro} and Theorem \\ref{strA} it has \nPBW-generators of the form (\\ref{vad1}):\n\\begin{equation}\nc_u=u^s+\\sum \\alpha _iW_i+\\ldots \\in \\hbox{\\bf U}, \\ \\ \\ u=u[k,m].\n\\label{vad25}\n\\end{equation}\nBy means of relations (\\ref{a1rel}) we have\n$p_{uu}=p(x_kx_{k+1}\\cdots x_m,$ $ x_kx_{k+1}\\cdots x_m)=q.$\nThus, if $q$ is not a root of 1, Lemma \\ref{nco1}\nshows that in (\\ref{vad25}) the exponent $s$ equals 1, \nwhile all heights of the $c_u$'s in {\\bf U} are infinite.\n\nIf $q$ has a finite multiplicative order $t>2,$ then $u[k,m]^t=0$ in $u_q^+(\\frak{ sl}_{n+1}),$\nsee for example \\cite[Theorem 3.2]{KA}; that is, by \\cite[Lemma 3.3]{KA}\nthe values of $u[k,m]$ are still the \nPBW-generators of $u_q^+(\\frak{ sl}_{n+1}),$ but all of them have the finite height $t.$\nBy Lemma \\ref{nco1} in (\\ref{vad25}) we have $s\\in \\{ 1, t, tl^r\\}.$ \nSince $u[k,m]^t=u[k,m]^{tl^r}=0,$ the exponent $s$\nin (\\ref{vad25}) equals 1, while all heights of the $c_u$'s in {\\bf U} equal $t.$\n\nHence in both cases the PBW-generators of {\\bf U} have the following form\n\\begin{equation}\nc_u=u[k,m]+\\sum \\alpha _iW_i+\\sum_j \\beta_jV_j \\in \\hbox{\\bf U}.\n\\label{vad22}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $u[k,m]$ super-letters, \n$D(W_i)=$ $D(u[k,m])$ $=x_k+x_{k+1}+\\ldots +x_m,$ and $V_j$ are $G$-super-words \nof $D$-degree less than $x_k+x_{k+1}+\\ldots +x_m.$\n\nNow, in order to reduce the freedom in construction of the PBW-generators, \nwe are going to show that (\\ref{vad22}) with homogeneous $c_u$ implies that \n{\\bf U} has an element of the same form that belongs to\na special finite set of elements (\\ref{cbr1}).\n\n\\begin{proposition} \nIf a right coideal subalgebra {\\bf U}$\\supseteq {\\bf k}[G]$ of $U_q^+(\\frak{ sl}_{n+1})$\nor $u_q^+(\\frak{ sl}_{n+1})$ contains a homogeneous element $c$\nwith the leading term $u[k,m],$ $k\\leq m,$ then for a suitable subset $\\bf S$\nof the interval $[k,m-1]$ the value of the below defined element\n$\\Psi ^{\\hbox{\\bf s}}(k,m)$ belongs to {\\bf U}.\n\\label{phi}\n\\end{proposition}\n\n\\begin{definition} \\rm\nLet {\\bf S} be a set of integers from the interval $[1,n].$ We define a\n{\\it piecewise continuous word related to} {\\bf S} as follows\n\\begin{equation}\nu^{\\hbox{\\bf s}}(k,m) \n\\stackrel{df}{=}u(1+s_r,m)u(1+s_{r-1},s_r)\\cdots u(1+s_1,s_2)u(k,s_1),\n\\label{pdw}\n\\end{equation}\nwhere {\\bf S}$\\, \\cap [k,m-1]=\\{ s_1,s_2,\\ldots s_r\\}, \\ $\n$k\\leq s_11.$\nThus by Lemma \\ref{ind} applied to (\\ref{cbr1}) the value in $U_q^+(\\frak{ sl}_{n+1})$\nor $u_q^+(\\frak{ sl}_{n+1})$ of the bracketing is independent\nof the alignment of the big brackets. In particular we have the following decomposition\n\\begin{equation}\n \\Psi ^{\\hbox{\\bf s}}(k,m)=\\hbox{\\Large [} \\Psi ^{\\hbox{\\bf s}}(1+s_i,m),\n\\Psi ^{\\hbox{\\bf s}}(k,s_i) \\hbox{\\Large ]}.\n\\label{cbrr}\n\\end{equation}\nOf course the word $u(k,m)$ is a piecewise continuous word with empty {\\bf S}, \nor more generally, with {\\bf S}$\\, \\cap [k,m-1]=\\emptyset .$\n\nLet us choose an arbitrary $s_{r+1},$ $s_r1+s_i; \\hfill \\cr\n0,\\hfill &\\hbox{if }s_{i+1}=1+s_i,\\hfill \\end{matrix} \\right.\n\\label{der7}\n\\end{equation}\nwhere \n$\\mu =(1-q^{-1})^2p(u(1+s_{i+1},m),x_{1+s_i}).$ \n\nFormulae (\\ref{der35} -- \\ref{der7}) show that products of pairwise separated\nelements from $W^{\\hbox{\\bf s}}(k,m)$ span a differential subspace, that contains\nall first derivatives of $\\Psi ^{\\hbox{\\bf s}}(k,m).$ Hence\nby induction it contains the derivatives of higher order as well. \n\nTo see that any product of pairwise separated\nelements from $W^{\\hbox{\\bf s}}(k,m)$ is proportional to some derivative\nof $\\Psi ^{\\hbox{\\bf s}}(k,m)$ we shall prove the following relation. \n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w=\\alpha \\in {\\bf k},\\, \\alpha \\neq 0,\\\n\\hbox{ if } \\ w=u^{\\hbox{\\bf s}}(k,m).\n\\label{der9}\n\\end{equation}\nIf {\\bf S}$\\, \\cap [k,m-1]=\\emptyset $ then $w=x_kx_{k+1}\\ldots x_m$ and relation follows from (\\ref{der1}).\nLet {\\bf S}$\\, \\cap [k,m-1]\\neq \\emptyset .$ By definition (\\ref{pdw}) we have $w=v\\cdot w^{\\prime }, $\n where $v=x_{1+s_r}x_{2+s_r}\\ldots x_m,$ $w^{\\prime }=u^{\\hbox{\\bf s}}(k,s_r).$\nHence \n$$\n\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w=\\partial_{1+s_r}(\\Psi ^{\\hbox{\\bf s}}(k,m))\n\\cdot D_{v^{\\prime }}D_{w^{\\prime }},\n$$\nwhere $v^{\\prime }=x_{2+s_r}\\ldots x_m.$ By (\\ref{der8})\nthe element $\\partial_{1+s_r}(\\Psi ^{\\hbox{\\bf s}}(k,m))$ is proportional\nto $u[2+s_r]\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_r).$ Since $\\Psi ^{\\hbox{\\bf s}}(k,s_r)$\nis independent of $x_j,$ $2+s_r\\leq j\\leq m,$ skew differential Leibniz rule (\\ref{defdif}) implies\n$$\n(u[2+s_r]\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_r))\\cdot D_{v^{\\prime }}=\n(u[2+s_r]\\cdot D_{v^{\\prime }})\\Psi ^{\\hbox{\\bf s}}(k,s_r).\n$$\nBy means of the multiple application of (\\ref{der1}) we see that \n$\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w$ is proportional to \n$\\Psi ^{\\hbox{\\bf s}}(k,s_r)\\cdot D_{w^{\\prime }}.$ By induction on $m-k$ \nwe get (\\ref{der9}).\n\n\\smallskip\nNow consider a product of separated elements from $W^{\\hbox{\\bf s}}(k,m),$\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}}(a_1,b_1)\\cdot \\Psi ^{\\hbox{\\bf s}}(a_2,b_2)\\ldots \\Psi ^{\\hbox{\\bf s}}(a_l,b_l),\n\\ \\ k\\leq b_ik.$ In this case by definition \n$a-1\\notin \\,${\\bf S}, say $s_i1+s_i.$ Hence by (\\ref{der7}),\nor by (\\ref{der4}) provided $i=0,$ or by (\\ref{der8}) provided $i=r$,\nthe element $\\partial _{1+s_i}(\\Psi ^{\\hbox{\\bf s}}(k,m))$ is proportional to \n$\\Psi ^{\\hbox{\\bf s}}(2+s_i,m)\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_i),$ where formally \n$\\Psi ^{\\hbox{\\bf s}}(k,s_0)=1.$ Since \n$\\Psi ^{\\hbox{\\bf s}}(2+s_i,m)$ is independent of $x_j,$ $k\\leq j\\leq s_i,$ formula\n(\\ref{der9}) shows that $\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_u,$ $u=x_{1+s_i}u^{\\hbox{\\bf s}}(k,s_i)$\nis proportional to $\\Psi ^{\\hbox{\\bf s}}(2+s_i,m).$ If $2+s_i=a,$ the required representation \nof $\\Psi ^{\\hbox{\\bf s}}(a,m)$ is found. If $2+s_i1.$ By definition $1+b_1s_{t+1}.$ Hence all terms in the sum (\\ref{s}),\nexcept one that corresponds \nto $i=s_{t+1},$ are zero. \nMoreover (\\ref{xy}) implies that, due to the choice of $T,$\nthe element $u^{[T]}(1+s_{t+1}, m)\\cdot D_w$ is a nonzero scalar,\nwhile $u^{[L]}(1+s_{t+1}, m)\\cdot D_w=0$ for any other super-word \n$u^{[L]}(1+s_{t+1}, m)$ that appears in the decomposition of $A_{s_{t+1}}$\nwith a nonzero coefficient.\nHence $A_{s_{t+1}}\\cdot D_w$ is a nonzero scalar $\\mu .$ Finally, we have\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1})=\\mu ^{-1} B\\cdot D_v \\in \\hbox{\\bf U}.\n\\label{pit2}\n\\end{equation}\n\n\n\\smallskip\n2. Let us derivate (\\ref{pit1}) by $x_{1+s_t}.$ By formulae \n(\\ref{der8}) and (\\ref{defdif}) we have\n$$\n\\mu u[(2+s_t, m)] \\cdot\n\\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\n$$\n\\begin{equation}\n+\\sum _{i=s_{t+1}}^{m-1}\\mu_iA_iu[(2+s_{t},i)]\\cdot \\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\\in \\, \\hbox{\\bf U},\n\\label{pit3}\n\\end{equation}\nwhere $\\mu =(1-q^{-1})^2,$ $\\mu_i=(1-q^{-1})^2p(A_i,x_{1+s_t})$ with the only possible \nexception\n$\\mu _{s_{t+1}}=(1-q^{-1})p(A_i,x_{1+s_t})$ provided $s_{t+1}=1+s_t.$\nHere we may apply (\\ref{der8}) \nsince the number $r$ related to {\\bf S}$_t$ \nequals $t.$\n\nDenote by $z$ the piecewise continuous word $u^{\\hbox{\\bf s}_{t-1}}(k,s_t).$\nLet us apply $\\cdot D_z$ to (\\ref{pit3}). Formula (\\ref{der9}) shows that\n$\\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\\cdot D_z$ is a nonzero scalar. Hence we get\n$$\n\\mu u[2+s_t,m]+\\sum _{i=s_{t+1}}^{m-1}\\mu_iA_iu[2+s_t,i]\\in \\hbox{\\bf U}.\n$$\nLet us apply $\\cdot D_w$ with $w=u(2+s_t,s_{t+1})$ to this sum.\nFormulae (\\ref{defdif}) and (\\ref{der4}) imply\n\\begin{equation}\n(1-q^{-1})u[1+s_{t+1},m]+\\beta _1A_{s_{t+1}}+\n(1-q^{-1})\\sum _{i>s_{t+1}}\\beta _iA_iu[1+s_{t+1},i]\\in \\hbox{\\bf U}.\n\\label{pit4}\n\\end{equation}\nwhere $\\beta _1=p(A_{s_{t+1}},x_{1+s_t}w)$ $=p_{vu},$ \n$\\beta _i$ $=p(A_i, x_{1+s_t}w)$ $=p_{v_{i}\\, u}$\nwith \n$$\nu=u(1+s_t,s_{t+1}), \\ v=u(1+s_{t+1},m),\\ v_i=u(1+i,m).\n$$\nLet us multiply the element (\\ref{pit4}) from the right by \n$\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1}) \\in \\, ${\\bf U}, see (\\ref{pit2}), and subtract the result from\n(\\ref{pit1}) multiplied by $\\beta _1.$ \nWe get\n$$\n\\beta _1\\Psi ^{\\hbox{\\bf s}_t}(k,m)+(q^{-1}-1)u[1+s_{t+1},m]\\cdot \n\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1})\n$$\n\\begin{equation}\n+\\sum _{i=s_{t+2}}^{m-1}A_i\\{\\beta _1 \\Psi ^{\\hbox{\\bf s}_t}(k,i)\n+(q^{-1}-1)\\beta_iu[1+s_{t+1},i]\\cdot \\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1}) \\} \\in \\hbox{\\bf U}.\n\\label{pit6}\n\\end{equation}\nBy the recurrence formula (\\ref{cbr3}) the first line of the above formula\nequals $-\\Psi ^{\\hbox{\\bf s}_{t+1}}(k,m),$ while the expression in the braces\nequals $-\\beta_i\\Psi^{\\hbox{\\bf s}_{t+1}}(k,i).$\nThus we get the required relation\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}_{t+1}}(k,m)+\\sum _{i=s_{t+2}}^{m-1} \n\\beta_iA_i\\Psi^{\\hbox{\\bf s}_{t+1}}(k,i)\\in \\hbox{\\bf U}.\n\\label{pit7}\n\\end{equation}\nProposition \\ref{phi} is proved.\n\n\\begin{corollary} \nIf $q$ is not a root of $1,$ then $U_q^+(\\frak{ sl}_{n+1})$ has\njust a finite number of right coideal subalgebras that include the coradical.\nIf the multiplicative order of $q$ equals $t>2,$ then \n$u_q^+(\\frak{ sl}_{n+1})$ has\njust a finite number of homogeneous right coideal subalgebras \nthat include the coradical.\n\\label{fin1}\n\\end{corollary}\n\\begin{proof}\nThis follows from Lemma \\ref{odn} and Propositions \\ref{pro}, \\ref{phi}.\n Indeed, one has $n(n-1)\/2$ options for possible value of an {\\bf U}-root\n(Definition \\ref{root}). There exists $2^{n(n-1)\/2}$ variants for sets of {\\bf U}-roots. For any given \nroot $\\gamma =x_k+x_{k+1}+\\ldots +x_m$ there exists not more than $2^{m-k}<2^n$\noptions for {\\bf S} to define a PBW-generator $\\Psi ^{\\hbox{\\bf s}}(k,m).$ Hence the total number of possible sets of PBW-generators is less than $n^{(2^n)}\\cdot 2^{n(n-1)\/2}.$ \\end{proof}\n\n\\section{Root sequence}\n\nOur next goal is to show that the exact number of (homogeneous) right coideal \nsubalgebras in $U_q^+(\\frak{ sl}_{n+1})$ (in $u_q^+(\\frak{ sl}_{n+1})$)\nthat contain {\\bf k}$\\, [G]$\nequals $(n+1)!.$ In what follows for short we shall denote by $[k:m]$ the element \n$x_k+x_{k+1}+\\ldots +x_m\\in \\Gamma ^+$ considered as an $U_q^+(\\frak{ sl}_{n+1})$-root.\n\n\\begin{definition} \\rm\nLet $\\gamma _k$ be a simple {\\bf U}-root of the form $[k:m]$ with the maximal $m.$\nDenote by $\\theta _k$ the number $m-k+1,$ the length (weight) of $\\gamma _k.$\nIf there are no simple {\\bf U}-roots of the form $[k:m],$ we put $\\theta _k=0.$\nThe sequence $r({\\bf U})=(\\theta_1, \\theta_2, \\ldots ,\\theta_n)$\nsatisfies $0\\leq \\theta_k\\leq n-k+1$ and it is uniquely defined by {\\bf U}.\nWe shall call $r({\\bf U})$ a {\\it root sequence of } {\\bf U}, or just an $r$-{\\it sequence of} {\\bf U}.\nBy $\\tilde{\\theta }_k$ we denote $k+\\theta_k -1,$ the maximal value of $m$ for the simple\n {\\bf U}-roots of the form $[k:m]$ with fixed $k.$\n\\label{tet}\n\\end{definition}\n\n\\begin{theorem} \nFor each sequence \n$\\theta=(\\theta_1, \\theta_2, \\ldots ,\\theta_n),$ such that $0\\leq \\theta_k\\leq n-k+1,$\n$1\\leq k\\leq n$ there exists one and only one $($homogeneous$)$ right coideal subalgebra\n{\\bf U}$\\, \\supseteq G$ of $U_q^+(\\frak{ sl}_{n+1})$ $($respectively, of $u_q^+(\\frak{ sl}_{n+1}))$\nwith $r({\\bf U})=\\theta .$ In what follows we shall denote this subalgebra by {\\bf U}$_{\\theta }.$\n\\label{teor}\n\\end{theorem}\nThe proof will result from the following lemmas.\n\n\\begin{lemma} \nIf $[k:m]$ is an {\\bf U}-root, then for each $r,$ $k\\leq r< m$\neither $[k:r]$ or $[r+1:m]$ is an {\\bf U}-root.\n\\label{su}\n\\end{lemma}\n\\begin{proof} By Proposition \\ref{phi} we have $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}\nfor a suitable {\\bf S}. If $r\\in \\, ${\\bf S}, then Theorem \\ref{26} and definition (\\ref{pbse}) imply\n$\\Psi ^{\\hbox{\\bf s}}(k,r)\\in \\, ${\\bf U}, hence $[k:r]$ is an {\\bf U}-root.\nIf $r\\notin \\, ${\\bf S}, then again Theorem \\ref{26} and (\\ref{pbse}) with $a=r+1$\nimply $\\Psi ^{\\hbox{\\bf s}}(r+1,m)\\in \\, ${\\bf U}, hence $[r+1:m]$ is an {\\bf U}-root.\n\\end{proof}\n\n\\begin{lemma} \nIf $[k:m]$ is a simple {\\bf U}-root, then \nthere exists only one subset {\\bf S} of the interval $[k,m-1],$\nsuch that $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}.\nMoreover the set {\\bf S} is uniquely defined by the set of all {\\bf U}-roots.\n\\label{su1}\n\\end{lemma}\n\\begin{proof} \nLet $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}.\nBy the definition of a simple root for each $r,$ $k\\leq r\\tilde{\\theta }_k$, while $[k,\\tilde{\\theta }_k]$ is a simple {\\bf U}-root. \n\nIf $m<\\tilde{\\theta }_k,$\nthen $[m+1:\\tilde{\\theta }_k]$ is an {\\bf U}-root if and only if it is a sum of simple {\\bf U}-roots\nstarting with a number greater than $k.$ Hence by induction the $r$-sequence\ndefines all roots of the form $[m+1:\\tilde{\\theta }_k],$ $k\\leq m<\\tilde{\\theta }_k.$ \n\nBy Lemma \\ref{su}\nthe weight $[k:m]$ is an {\\bf U}-root if and only if $[m+1: \\tilde{\\theta }_k]$ is not an {\\bf U}-root\n(recall that $[k:\\tilde{\\theta }_k]$ is simple). Hence the $r$-sequence also defines the set of\nall {\\bf U}-roots of the form $[k:m],$ $m<\\tilde{\\theta }_k.$ An {\\bf U}-root $[k:m],$\n$m<\\tilde{\\theta }_k$\nis simple if and only if there does not exist $r,$ $k\\leq rk$ implies $m\\in T_{k_1}.$\nAt the same time $k_1-1\\in R_k,$ hence by definition $m\\in T_k.$\n\n\\smallskip\n\\noindent\n{\\sc Claim} 2. {\\it If $s\\in T_k,$ $m\\in T_{s+1},$ then $m\\in T_k.$} \n\n\\smallskip\n\\noindent\nBy means of Claim 1 applied to $s$ we find a sequence \n$k_0=kk.$ Conditions (\\ref{pet1}$a$) and (\\ref{pet1}$b$) are valid for $m\\leftarrow s.$\nSuppose that (\\ref{pet1}$c$) fails. In this case we may find a number $t,$ $k\\leq t1.$ Again by the first\nclaim we have $m\\in T_{k_1}.$\nSince $k_1-1$ belongs to $R_k,$ it satisfies condition (\\ref{pet1}$b$), \n$\\tilde{\\theta }_k\\notin T_{k_1}.$ However Claim 2 shows that the conditions \n $m\\in T_{k_1},$ $\\tilde{\\theta }_k\\in T_{m+1} $\n imply $\\tilde{\\theta }_k\\in T_{k_1}.$\nA contradiction, that proves the claim.\n\n\\smallskip\n\\noindent\n{\\sc Claim 5}. {\\it The subalgebra $U^{\\prime }$ generated by $\\Psi ^{\\hbox{\\bf s}}(k,m),$\n$1\\leq k\\leq m\\leq n,$ $m\\in T_k,$ {\\bf S}$\\, =T_k$ is a differential subalgebra.}\n\n\\smallskip\n\\noindent\nIt suffices to show that all partial derivatives of $\\Psi ^{\\hbox{\\bf s}}(k,m)$ belong to $U.$\nBy Theorem \\ref{26} we have to check that $\\Psi ^{T_k}(a,b)\\in U$ provided that\n$b\\in T_k,$ $a-1\\notin T_k,$ $k\\leq a\\leq b\\leq m.$ By definition\n$\\Psi ^{T_a}(a,b)\\in U$ since due to the third claim $b\\in T_a.$ If \n\\begin{equation}\nT_k\\cap [a, b-1]=T_a\\cap [a,b-1],\n\\label{tor}\n\\end{equation}\nthen we have nothing to prove. In general, however, just the inclusion\n$T_k\\cap [a, b-1]\\subseteq T_a\\cap [a,b-1]$ holds: if $t\\in T_k,$ $a\\leq t,$\nthen Claim 3 with $s\\leftarrow a-1$ says $t\\in T_a$ (since $a-1\\notin T_k$).\n\nWe shall prove $\\Psi ^{T_k}(a,b)\\in U$ by induction on $b-a.$ If $b=a,$\nthen (\\ref{tor}) certainly holds.\n\nLet us choose the minimal $s\\in T_a,$ $s\\notin T_k.$ Then $T_k\\cap [a, s-1]=T_a\\cap [a,s-1].$\nHence $\\Psi ^{T_k}(a,s)=\\Psi ^{T_a}(a,s)\\in U.$\nBy the inductive supposition applied to the interval $[s+1,b]$\nwe get $\\Psi ^{T_k}(s+1,b)\\in U.$\nBy decomposition (\\ref{cbrr}) we have \n$$\n\\Psi ^{{T_k}\\cup \\{ s \\}}(a,b)=[\\Psi ^{T_k}(s+1,b), \\Psi ^{T_k}(a,s)]\\in U.\n$$\nAt the same time (\\ref{cby}) implies\n$$\n\\Psi ^{{T_k}\\cup \\{ s \\}}(a,b)-(1-q^{-1})\\Psi ^{T_k}(s+1,b)\\cdot \\Psi ^{T_k}(a,s)=\n-p_{vu}\\Psi ^{T_k}(a,b).\n$$\nTherefore $\\Psi ^{T_k}(a,b)\\in U,$ which is required.\n\n\\smallskip\n\\noindent\n{\\sc Claim 6}. {\\it {\\bf U}$\\, =U\\#{\\bf k}[G]$ is a right coideal subalgebra.}\n\n\\smallskip\n\\noindent\nSince $U$ is homogeneous in each variable, we have \n$g^{-1}Ug\\subseteq U,$ $g\\in G.$\nIt remains to apply Lemma \\ref{qsim}.\n\n\\smallskip\n\\noindent\n{\\sc Claim 7}. {\\it The set of all \\, {\\bf U}-roots is $\\{ [k:m]\\, |\\, m\\in T_k\\}.$ In particular\n$\\{ \\Psi ^{T_k}(k,m)\\, |$ $m\\in T_k\\} $ is a set of PBW-generators of {\\bf U} over {\\bf k}$[G].$}\n\n\\smallskip\n\\noindent\nIf $\\gamma =[a:b]$ is an {\\bf U}-root, then, by definition, in {\\bf U} there exists\na homogeneous element (\\ref{vad22}) of degree $\\gamma .$ Since by definition \n$U$ is generated by $\\{ \\Psi ^{T_k}(k,m)\\, |\\, m\\in T_k\\} ,$ the degree $\\gamma $\nis a sum of degrees of the generators: \n$\\gamma =[k_1:k_2-1]+[k_2:k_3-1]+\\ldots +[k_{r-1}:k_r-1],$ $k_{i+1}-1\\in T_{k_i},$ $1\\leq i2$) that do not contain the coradical. First of all we note that for every submonoid \n$\\Omega \\subseteq G$ the set of all linear combinations {\\bf k}$\\, [\\Omega]$\nis a right coideal subalgebra.\nConversely if $U_0\\subseteq \\,${\\bf k}$\\, [G]$ is a right coideal subalgebra then\n$U_0=\\, ${\\bf k}$\\, [\\Omega]$ for $\\Omega =U_0\\cap G$\nsince $a=\\sum_i \\alpha_i g_i\\in U_0$ implies $\\Delta (a)=\n\\sum_i \\alpha_i g_i\\otimes g_i\\in U_0\\otimes \\, ${\\bf k}$\\, [G];$ that is, $\\alpha_i g_i\\in U_0.$\n\\begin{definition} \\rm\nFor a sequence $\\theta=(\\theta_1, \\theta_2, \\ldots ,\\theta_n),$ such that $0\\leq \\theta_k\\leq n-k+1,$\n$1\\leq k\\leq n$ we denote by {\\bf U}$^1_{\\theta }$ a subalgebra with 1 generated by \n$g^{-1}\\Psi ^{\\hbox{\\bf s}}(k,m),$ where $g=g_kg_{k+1}\\ldots g_m,$ and $\\Psi ^{\\hbox{\\bf s}}(k,m)$\nruns through the set of PBW-generators of {\\bf U}$_{\\theta },$ see Theorem \\ref{teor}\nand Claim 7, Section 5.\n\\label{1te}\n\\end{definition}\n\\begin{lemma} The subalgebra\n{\\bf U}$^1_{\\theta }$ is a homogeneous right coideal, and \n{\\bf U}$^1_{\\theta }\\cap G=\\{ 1\\} .$ \n\\label{1su}\n\\end{lemma}\n\\begin{proof}\nThe subalgebra\n{\\bf U}$^1_{\\theta }$ is homogeneous since it is generated by homogeneous elements.\nIts zero homogeneous component equals {\\bf k} since among the generators just one,\nthe unity, has zero degree. \n\nDenote by $A_{\\theta }$ a {\\bf k}-subalgebra generated by \nthe PBW-generators $\\Psi ^{\\hbox{\\bf s}}(k,m)$ of {\\bf U}$_{\\theta }.$\nThe algebra {\\bf U}$^1_{\\theta }$ is spanned by all elements of the form \n$g_a^{-1}a,$ $a\\in A_{\\theta }.$ Since {\\bf U}$_{\\theta }$ is a right coideal, \nfor any homogeneous $a\\in A_{\\theta }$ we have\n$\\Delta (a)=\\sum g(a^{(2)})a^{(1)}\\otimes a^{(2)}$ where $a^{(1)}\\in A_{\\theta },$\n$g_a=g(a^{(1)})g(a^{(2)}).$ Therefore \n$\\Delta (g_a^{-1}a)=\\sum g(a^{(1)})^{-1}a^{(1)}\\otimes g_a^{-1}a^{(2)}$ with \n$g(a^{(1)})^{-1}a^{(1)}\\in \\, ${\\bf U}$_{\\theta }^1.$\n\\end{proof}\n\\begin{theorem} If $U$ is a homogeneous right coideal subalgebra \nof $U^{+}_q(\\mathfrak{sl}_{n+1})\\, ($resp. of $u^{+}_q(\\mathfrak{sl}_{n+1}))$\nsuch that $\\Omega \\stackrel{df}{=} U\\cap G$ is a group,\nthen $U=\\, ${\\bf U}$_{\\theta }^{1}\\, ${\\bf k }$[\\Omega ]$\nfor a suitable $ \\theta .$\n\\label{orc}\n\\end{theorem}\n\\begin{proof}\nLet $u=\\sum h_ia_i\\in U$ be a homogeneous element of degree $\\gamma \\in \\Gamma ^{+}$\nwith different $h_i\\in G,$ and $a_i\\in A,$ where by $A$ we denote the {\\bf k}-subalgebra\ngenerated by $x_i,$ $1\\leq i\\leq n.$ Denote by ${\\pi }_{\\gamma }$ the natural\nprojection on the homogeneous component of degree $\\gamma .$\nRespectively $\\pi _g,$ $g\\in G$ is a projection on the subspace {\\bf k}$\\, g.$\nWe have $\\Delta (u)\\cdot (\\pi _{\\gamma }\\otimes \\pi _{h_i})=h_ia_i\\otimes h_i.$\nThus $h_ia_i\\in U.$\n\nBy Theorem \\ref{teor} we have {\\bf k}$\\, [G]U=\\,${\\bf U}$_{\\theta }$ for a suitable $\\theta .$\nIf $u=ha\\in U,$ $h\\in G,$ $a\\in A,$ then $\\Delta (u)\\cdot (\\pi _{hg_a}\\otimes \\pi _{\\gamma })=\nhg_a\\otimes ha.$ Therefore $hg_a\\in U\\cap G=\\Omega ;$ that is, $u=\\omega g_a^{-1}a,$\n$\\omega \\in \\Omega .$ Since $\\Omega $ is a subgroup we get $g_a^{-1}a\\in U.$\nIt remains to note that all elements $g_a^{-1}a,$ such that $ha\\in U$ span the algebra \n{\\bf U}$_{\\theta }^1.$\n\\end{proof}\nIf $U\\cap G$ is not a group then $U$ may have a more complicated structure.\n\\begin{example} \\rm \nLet $\\Omega $ be a submonoid of $G.$ Denote by $\\overline{\\Omega }$ an arbitrary family of sets \n$\\{ \\Omega _{\\gamma }, \\gamma \\in \\Gamma ^{+}\\} $ that satisfies \nthe following conditions\n$$\n\\Omega _0=\\Omega, \\ \\ \\ \\ \n\\Omega _{\\gamma }\\cdot \\Omega _{\\gamma ^{\\prime }}\\subseteq \n\\Omega _{\\gamma +\\gamma ^{\\prime }}\\subseteq\n \\Omega _{\\gamma }\\cap \\Omega _{\\gamma ^{\\prime }}.\n$$\nIn this case the linear space {\\bf U}$_{\\theta }^{\\overline{\\Omega }}$ spanned\nby the elements $\\omega _{\\gamma }a,$ \n$\\omega _{\\gamma }\\in \\Omega _{\\gamma },$ $a\\in \\, ${\\bf U}$_{\\theta }^1,$\ndeg$\\, (a)=\\gamma $ is a right coideal subalgebra such that \n{\\bf U}$_{\\theta }^{\\overline{\\Omega }}\\cap G=\\Omega .$ The $\\gamma $-homogeneous component\nof this algebra equals $\\Omega _{\\gamma }\\, (${\\bf U}$_{\\theta }^1)_{\\gamma }.$\n Hence different $\\overline{\\Omega }$ define different homogeneous right coideal subalgebras.\n\\label{oxc1}\n\\end{example}\nFinally we point out a simplest one-parameter family of inhomogeneous \nright coideal subalgebras that have trivial intersection with the coradical.\n\\begin{example} \\rm \nLet $a=g_1^{-1}(x_1+\\alpha ),$ $\\alpha \\in \\,${\\bf k}.\nWe have \n$$\n\\Delta (a)=g_1^{-1}x_1\\otimes g_1^{-1}+ 1\\otimes g_1^{-1}x_1+\\alpha g_1^{-1}\\otimes g_1^{-1}\n=a\\otimes g_1^{-1}+1\\otimes g_1^{-1}x_1.\n$$\nTherefore the two-dimensional space spanned by $a$ and $1$ is a right coideal. \nThus the algebra {\\bf k}$\\, [a]$ with 1 generated by $a$ is a right coideal subalgebra,\nin which case {\\bf k}$\\, [a]\\cap G=\\{ 1\\} .$\n\\label{oxc2}\n\\end{example}\n\n\\section{K\\'eb\\'e construction and ${\\rm ad}_r$-invariant subalgebras}\n\nIn this section we characterize ad$_r$-invariant right coideal subalgebras that have trivial intersection \nwith the coradical in terms of K\\'eb\\'e's construction \\cite{Keb, Keb1}. Recall that the right \nadjoint action of a Hopf algebra $H$ on itself is defined by the formula\n$$\n({\\rm ad}_ra)b=\\sum \\sigma (a^{(1)})ba^{(2)},\n$$\nwhere $\\sigma $ is the antipode. The map $a\\rightarrow \\, $ad$_ra$ is a homomorphism \nof algebras ad$_r:H\\rightarrow \\hbox{End}H.$ In particular a subspace is invariant under \nthe action of all operators ad$_rH$ if and only if it is invariant under the actions of ad$_rh_i$\nfor some set of generators $\\{ h_i\\}.$ For $H=U_q^+(\\mathfrak{sl}_{n+1})$ or for\n$H=u_q^+(\\mathfrak{sl}_{n+1})$ we have\n$$\n(\\hbox{ad}_rg)b=g^{-1}bg, \\ \\ g\\in G; \\ \\ \\ (\\hbox{ad}_rx_i)b=g_i^{-1}(bx_i-x_ib).\n$$\nThe latter equality would be more familiar if we take $b=g_a^{-1}a$ with\n$a\\in A\\stackrel{df}{=}\\hbox{\\bf k}\\langle x_1,\\ldots ,x_n\\rangle :$\n\\begin{equation}\n(\\hbox{ad}_rx_i)(g_a^{-1}a)=g_i^{-1}(g_a^{-1}ax_i-x_ig_a^{-1}a)=-g_i^{-1}g_a^{-1}[x_i,a].\n\\label{keb1}\n\\end{equation}\nIn particular the subalgebra $H^1$ generated by $g_a^{-1}a,$ $a\\in A$ \n(in our terms this is {\\bf U}$^1_{\\theta }$ for $\\theta =(1,1,\\ldots, 1)$) is ad$_r$-invariant.\n\n\nThe following construction of ad$_r$-invariant right coideal subalgebras appeared in \n\\cite{Keb, Keb1}, see also \\cite[Section 6]{Let}. Let $\\pi $ be a subset of $[1,k].$\nDenote by $K(\\pi )$ a subalgebra generated by elements of the form\n$$\n\\hbox{ad}_r(x_{i_1}x_{i_2}\\ldots x_{i_k})\\, g_j^{-1}x_j, \\ \\ \\ j\\in \\pi,\\ \\ i_r\\notin \\pi, 1\\leq r\\leq k.\n$$\nThe algebra $K(\\pi )$ is ad$_r$-invariant right coideal (see, \\cite[Lemma 1.2]{Let} up to a left-right symmetry). This is homogeneous, and $K(\\pi )\\cap G=\\{ 1\\}$\nsince due to (\\ref{keb1}) the inclusion $K(\\pi )\\subseteq H^1$ is valid. Thus by Theorem \\ref{orc} \nwe have $K(\\pi )={\\bf U}^1_{\\theta }$ for a suitable $\\theta .$\n \n\\begin{theorem}\nThe following conditions on $U=\\,${\\bf U}$^1_{\\theta }$ are equivalent\n\ni. $U$ is {\\rm ad}$_r$-invariant.\n\nii. The sets $T_k,$ see Definition $\\ref{tski},$ have the form $T_k=[j(k),n],$ where \n$$\nj(k)\\stackrel{df}{=}\\, {\\rm min}\\, \\{ j\\, |\\, k\\leq j,\\ \\ j\\in T_j\\} .\n$$\n\niii. $U=K(\\pi )$ for a suitable $\\pi \\subseteq [1,n].$ \n\\label{Korc}\n\\end{theorem}\n\\begin{proof}\n{\\it iii}$\\, \\Rightarrow \\,${\\it i}. We have mentioned above.\n \n{\\it i}$\\, \\Rightarrow \\,${\\it ii}. By Claim 7 we have $m\\in T_k$ if and only if \n$\\Psi ^{T_k}(k,m)\\in \\hbox{\\bf U}_{\\theta }.$ In particular $j\\in T_j$ if and only if \n$x_j\\in \\hbox{\\bf U}_{\\theta },$ or, equivalently, $g_j^{-1}x_j\\in U.$\nIf $j\\in T_j$ and $k\\leq j,$ then by (\\ref{keb1}) we have\n$$\n{\\rm ad}_r(x_{j-1}x_{j-2}\\ldots x_k)\\ g_j^{-1}x_j=g_{u[k,j]}^{-1}u[k,j]\\in U.\n$$\nHence, by definition $[k:j]$ is an {\\bf U}$_{\\theta }$-root; that is, \naccording to Claim 7, we get $j\\in T_k.$ Moreover, if $i>j$ then by\n(\\ref{cin}) we have \n$$\n{\\rm ad}_r(x_{j+1}x_{j+2}\\ldots x_i)\\ g_{u[k,j]}^{-1}u[k,j]=g^{-1}[x_i,[x_{i-1},\\ldots [x_{j+1}, u[k,j]]\\ldots]]\n$$\n$$\n\\sim g^{-1}\\Psi ^{\\{ j+1,j+2,\\ldots ,i\\}}(k,i),\n$$\nwhere $g=g_kg_{k+1}\\ldots g_i.$ In particular $[k:i]$ is an {\\bf U}$_{\\theta }$-root; that is, \nagain according to Claim 7, we get $i\\in T_k.$ This proves $[j(k),n]\\subseteq T_k.$\n\nIf $m$ is the smallest element from $T_k$ then $u[k,m]=\\Psi ^{T_k}(k,m)\\in \\hbox{\\bf U}_{\\theta },$\nhence by multiple use of (\\ref{der1}) we get $x_m\\in \\hbox{\\bf U}_{\\theta };$ that is, $m\\in T_m,$\nand $m=j(k).$\n\n{\\it ii}$\\, \\Rightarrow \\,${\\it iii}. Let $\\pi =\\{ j\\, |\\, j\\in T_j\\} .$ For all $k,m,j$ such that \n$m\\in T_k,$ $j=j(k)$ we have \n$$\n{\\rm ad}_r(x_{j-1}x_{j-2}\\ldots x_kx_{j+1}x_{j+2}\\ldots x_m)\\ g_j^{-1}x_j\n$$\n$$=g^{-1}[x_m,[x_{m-1},\\ldots [x_{j+1}, u[k,j]]\\ldots]]=g^{-1}\\Psi ^{T_k}(k,m),\n$$\nwhere $g=g_mg_{m-1}\\ldots g_k.$ Since $K(\\pi )$ is ad$_r$-invariant, we get\n$g^{-1}\\Psi ^{T_k}(k,m)\\in K(\\pi ).$ Now Definition \\ref{1te} implies $U\\subseteq K(\\pi ).$\n\nSince $g_j^{-1}x_j\\in U,$ to check $K(\\pi )\\subseteq U$ it remains to show that\nad$_r(x_i)U\\subseteq U$ for $i\\notin \\pi .$ By (\\ref{br1}) and (\\ref{keb1}) it suffices to prove that \n$[x_i , \\Psi ^{T_k}(k,m)]\\in {\\bf U}_{\\theta }$ for $i\\notin \\pi ,$ $m\\in T_k.$\n\nLet $i=k-1.$ Since $k-1=i\\notin \\pi ,$ we have $k-1\\notin T_{k-1},$ and hence \n$j(k-1)=j(k).$ Eq. (\\ref{cbry})\nimplies $[x_{k-1}, \\Psi ^{T_k}(k,m)]\\sim \\Psi ^{T_{k-1}}(k-1,m)\\in {\\bf U}_{\\theta },$\nwhere $m\\in T_{k-1}$ follows from $T_{k-1}=[j(k),n]=T_k.$\n\nIf $ij(k),$ and $m+1\\in T_k.$ Now formula (\\ref{cin}) yields \n$$[x_{m+1}, \\Psi ^{T_k}(k,m)]\\sim \\Psi ^{T_k}(k,m+1)\\in {\\bf U}_{\\theta }.$$\n\nIf $i>m+1,$ then $[x_i , \\Psi ^{T_k}(k,m)]=0$ since $x_i$ and $\\Psi ^{T_k}(k,m)$\nare separated.\n\nWe shall show by induction on $m-k$ that in all remaining cases $[x_i , \\Psi ^{T_k}(k,m)]=0.$\nMore precisely, we prove $[x_i, \\Psi ^{\\hbox{\\bf s}}(k,m)]=0$ provided that \n{\\bf S} has the form $[j,n],$ and $k\\leq i\\leq m,$ $i\\neq \\hbox{\\rm min}\\, \\{ j,m \\}.$ \n\nIf $m-k=1,$ then for $j\\geq m$ we have just one option $i=k.$ The required relation \ntakes the form $[x_k,[x_k,x_{k+1}]]=0$ which is one of the defining relations (\\ref{rela}).\nFor $j=k$ we also have just one option $i=m=k+1.$ The required relation is \n$[x_m,[x_m,x_{m-1}]]=0.$ This relation is valid in $U_q^{+}(\\mathfrak{sl}_{n+1})$\nsince (\\ref{a1rel}) imply $[x_m,[x_m,x_{m-1}]]\\sim [[x_{m-1},x_m],x_m],$\nsee, for example, \\cite[Corollary 4.10]{Kh4}.\n\nIf $m-k>2$ then either $ik+1.$ In the former case we have $[x_i,x_m]=0,$\nand by the inductive supposition $[x_i , \\Psi ^{\\hbox{\\bf s}}(k,m-1)]=0.$ Hence representation\n(\\ref{cin}) implies the required equality. In the latter case we have $[x_i,x_k]=0,$\nand by the inductive supposition $[x_i , \\Psi ^{\\hbox{\\bf s}}(k+1,m)]=0.$ In this case representation\n(\\ref{cin1}) implies the required equality.\n\nFinally, suppose that $m-k=2.$ To simplify the notations we put $k=1,$ $m=3.$\n\nIf $j\\geq 3$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[[x_1,x_2],x_3],$ \nand we have two options $i=1,$ $i=2.$ If $i=1,$ we have to show\n$[x_1,[[x_1,x_2],x_3]]=0.$ This relation is valid since $x_1$ (skew)commutes both with \n$[x_1,x_2]$ and $x_3$ (but not vise versa: $[x_1,x_2]$ does not (skew)commute with $x_1$\nsince $[[x_1,x_2],x_1]\\neq 0!$) Let $i=2.$ We may apply (\\ref{bri}) since \n$p_{21}p_{22}p_{23}\\cdot p_{12}p_{22}p_{32}=1.$ Thus \nby (\\ref{bri}) and (\\ref{jak3}) we have \n$$\n[x_2,[[x_1,x_2],x_3]]\\sim [[[x_1,x_2],x_3],x_2]=[[x_1,[x_2,x_3]],x_2].\n$$\nThe word $x_1x_2x_3x_2$ is standard, and the standard alignment of brackets is precisely\n$[[x_1,[x_2,x_3]],x_2].$ Hence by the third statement of Theorem \\ref{strA} this is zero in \n$U_q^{+}(\\mathfrak{sl}_{n+1}).$\n\nIf $j=2,$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[x_3, [x_1,x_2]],$ and \nwe have two options $i=1,$ $i=3.$ If $i=1$ then\n$[x_1,[x_3,[x_1,x_2]]]=0$ since $x_1$ (skew)commutes both with \n$[x_1,x_2]$ and $x_3.$ Let $i=3.$ By (\\ref{jak4}) we have $[x_3,[x_1,x_2]]\\sim [x_1,[x_3,x_2]]].$\nSince $x_3$ (skew)commutes both with $x_1$ and $[x_3,x_2],$ we get \n$[x_3,[x_1,[x_3,x_2]]]=0.$\n\nIf $j=1$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[[x_3,x_2],x_1],$\n and we have two options $i=2,$ $i=3.$ If $i=3$ then\n$[x_3,[[x_3,x_2],x_1]]=0$ since $x_3$ (skew)commutes both with \n$[x_3,x_2]$ and $x_1.$ Let $i=2.$ \nWe may use (\\ref{bri}) since \n$p_{23}p_{22}p_{21}\\cdot p_{32}p_{22}p_{21}=1.$\nThus by (\\ref{bri}) and (\\ref{jak3}) we have \n$$\n[x_2,[[x_3,x_2],x_1]]\\sim [[[x_3,x_2],x_1],x_2]=[[x_3,[x_2,x_1]],x_2].\n$$\nThis element in new variables $y_1=x_3,$ $y_2=x_2,$ $y_3=x_3$ takes up the form\n$[[y_1,[y_2,y_3]],y_2].$ By the third statement of Theorem \\ref{strA}\nthis is zero in $U_q^{+}(\\mathfrak{sl}_{n+1}).$\n\\end{proof}\n\n\\section{Examples}\nIn this section by means of Theorem \\ref{teor} we provide some examples of right coideal subalgebras in $U_q^+(sl_n)$ or $u_q^+(sl_n)$ with their main characteristics:\nPBW-generators, the root sequence $r({\\bf U}),$ \nthe sets $T_i,$ $R_i,$ right coideal subalgebra generators, and maximal Hopf subalgebras.\nWe start with a characterization of \n$2^n$ ``trivial\" examples --- Hopf subalgebras.\n\n\\begin{proposition} \nA right coideal subalgebra {\\bf U}$={\\bf U}_{\\theta }$\nis a Hopf subalgebra if and only if \nfor every $k,$ $1\\leq k\\leq n$ either $\\theta _k=0$ or $\\theta _k=1.$\nAn algebra {\\bf U}$_{\\theta },$ with $\\theta _i\\leq 1$\nis generated over {\\bf k}$[G]$ by all $x_k$ with $\\theta _k=1.$ \n\\label{hop}\n\\end{proposition}\n\\begin{proof} \nIf $\\theta _k\\leq 1,$ $1\\leq k\\leq n,$ then Definition \\ref{pet1} shows that $R_k=\\{ k\\} $\nprovided that $\\theta _k=1$ and $R_k=\\emptyset $ otherwise.\nHence by Claim 8 the algebra {\\bf U} is generated over {\\bf k}$[G]$ \nby all $x_k$ with $\\theta _k=1.$ In particular {\\bf U} is a Hopf subalgebra of\n$U_q^+(sl_n).$ \n\nConversely, let {\\bf U} be a Hopf subalgebra. According to Claim 8 the algebra {\\bf U} is generated \nover {\\bf k}$[G]$ by the elements $a$ of the form $\\Psi ^{T_k}(k,m)$ \nwith $[k:m]$ being the simple {\\bf U}-roots.\nWe have $\\Delta (a)=\\sum a^{(1)}\\otimes a^{(2)}$ with \n$a^{(1)}, a^{(2)}\\in \\, ${\\bf U}. Since $[k:m]$ $=D(a)$ $=D(a^{(1)})+D(a^{(2)})$ and $[k:m]$ is simple,\nwe have either $D(a^{(1)})=0,$ or $D(a^{(2)})=0.$ Thus $a$ is a skew primitive element;\nthat is, $a=x_k$ is the only option for $a$ (see the second statement of Theorem \\ref{strA} for $q^t\\neq1,$ and comments after that theorem for $q^t=1).$\nIn particular all simple roots are of length 1, while Definition \\ref{tet} implies \n$\\theta _k\\leq 1.$ \\end{proof}\n\nNow we consider three special cases.\n\n\\begin{example} \\rm \nConsider the root sequence with the maximal possible components,\n$r({\\bf U})=(n,n-1,n-2,\\ldots ,2,1).$\n In this case by definition $T_n=R_n=\\{ n\\}.$ For $k2,$ \nthen this is the case for $\\Gamma $-homogeneous\nright coideal subalgebras of $u_q(\\frak{ sl}_{n+1}).$\n\\label{raz2}\n\\end{lemma}\n\\begin{proof} \nDue to the triangular decompositions (\\ref{tr}), (\\ref{tr1})\nthe values of super-letters $[x_kx_{k+1}\\ldots x_m],$ $[x_k^-x_{k+1}^-\\ldots x_m^-]$\nform a set of PBW-generators over {\\bf k}$[H]$ for $U_q(\\frak{ sl}_{n+1}).$ \n\nLet us fix the following order on the skew-primitive generators\n\\begin{equation} \nx_1>x_2>\\ldots >x_n>x_1^->x_2^->\\ldots >x_n^-.\n\\label{orr}\n\\end{equation}\nBy Lemma \\ref{nco1} and Proposition \\ref{pro} (see the arguments above Eq. (\\ref{vad22}))\nthe subalgebra {\\bf U} has PBW-generators of the form\n\\begin{equation}\nc=[u]+\\sum \\alpha _iW_i+\\sum_j\\beta_jV_j \\in \\hbox{\\bf U},\n\\label{vad10}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $[u]$ super-letters, \n$D(W_i)=D(u),$ and $V_j$ are $G$-super-words of $D$-degree less than $D(u),$ while\nthe leading term $[u]$ equals either $[x_kx_{k+1}\\ldots x_m]$ or \n$[x_k^-x_{k+1}^-\\ldots x_m^-].$ \nCertainly the leading terms here are defined by the degree function into the\nadditive monoid $\\Gamma ^+\\oplus \\Gamma ^-$ generated by $X\\cup X^-$\n(but not into the group $\\Gamma $!).\nIn particular all $W_i$ in (\\ref{vad10}) have the same constitution in \n$X\\cup X^-$ as the leading term $[u]$ does. Thus all $W_i$'s and \nthe leading term $[u]$\nbelong to the same component of the triangular decomposition. Hence\nit remains to show that there are no terms $V_j.$\n\nIf $q$ is not a root of 1 then\nby Corollary \\ref{odn11} the algebra {\\bf U} is $\\Gamma $-homogeneous.\nHence (in both cases) the PBW-generators may be chosen to be $\\Gamma $-homogeneous as well. \nIn this case all terms $V_j$ have the same $\\Gamma $-degree and smaller \n$\\Gamma ^+\\oplus \\Gamma ^-$-degree. However this is impossible.\n\nIndeed, if the leading term is $[x_k^-x_{k+1}^-\\ldots x_m^-]$ \nthen the $\\Gamma ^+\\oplus \\Gamma ^-$-degree\nof $V_j$ should be less than $x_k^-+x_{k+1}^-+\\ldots +x_m^-.$ Hence due to\ndefinitions (\\ref{orr}) and (\\ref{ord}) we have $V_j\\in U_q^-(\\frak{ sl}_{n+1}),$\n(respectively, $V_j\\in u_q^-(\\frak{ sl}_{n+1})$),\nand the $\\Gamma $-degree of $V_j$ coincides with the $\\Gamma ^+\\oplus \\Gamma ^-$-degree.\nA contradiction.\n\nSuppose that the leading term is $[x_k^+x_{k+1}^+\\ldots x_m^+].$\n Let $d=\\sum s_ix_i+\\sum r_ix_i^-$\nbe the $\\Gamma ^+\\oplus \\Gamma ^-$-degree of $V_j.$ Since \n\\begin{equation} \nd2,$ then\n$u_q(\\frak{ sl}_{n+1})$ has\njust a finite number of $\\Gamma $-homogeneous \nright coideal subalgebras containing the coradical.\n\\label{fin2}\n\\end{corollary}\n\\begin{proof}\nThis follows from the above lemma and Corollary \\ref{fin1} applied to $U_q^{\\pm }(\\frak{ sl}_{n+1}),$\n$u_q^{\\pm }(\\frak{ sl}_{n+1}).$\n \\end{proof}\n\nOur next goal is to understand when tensor product (\\ref{tru}) is a subalgebra\nand then to find a way to calculate the total number of ($\\Gamma $-homogeneous)\nright coideal subalgebras.\n\\begin{lemma} The tensor product $(\\ref{tru})$\nis a right coideal subalgebra if and only if \n\\begin{equation}\n [{\\bf U}^+,{\\bf U}^-]\\subseteq \\, {\\bf U}^-\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]} {\\bf U}^+.\n\\label{rud}\n\\end{equation}\n\\label{raz1}\n\\end{lemma}\n\\begin{proof} Of course if {\\bf U} is a subalgebra then (\\ref{rud}) holds.\nConversely, it is clear that {\\bf U} is a right coideal. Relation (\\ref{rud})\nimplies $u^+\\cdot v^-$ $=[u^+, v^-]$ $+p(u^+,v^-) v^-\\cdot u^+$ $\\in \\, ${\\bf U}, \nwhere $u^+\\in {\\bf U}^+,$ $v^-\\in {\\bf U}^-.$\nHence $(u^-\\cdot u^+)(v^-\\cdot v^+)$ $=u^-(u^+\\cdot v^-)v^+$\n$\\in \\,${\\bf U}, with arbitrary $v^+\\in {\\bf U}^+,$ $u^-\\in {\\bf U}^-.$\n\nSince ${\\bf U}={\\bf U}^-\\cdot H\\cdot {\\bf U}^+,$\nit remains to check that ${\\bf U}^-\\cdot H$ $=H\\cdot {\\bf U}^-,$ and\n${\\bf U}^+\\cdot H$ $=H\\cdot {\\bf U}^+.$ \nSince ${\\bf U}^+$ contains $G,$ it is homogeneous with respect to the grading \n(\\ref{grad}). If $u\\in ({\\bf U}^+)^{\\chi },$ $f\\in F,$ then $uf=\\chi (f)fu.$\nHence ${\\bf U}^+\\cdot F=F\\cdot {\\bf U}^+.$ Similarly \n${\\bf U}^-\\cdot G=G\\cdot {\\bf U}^-.$ \\end{proof}\n\n\n\\section{Consistency condition}\nIn this section we are going to find sufficient condition for consistency relation\n (\\ref{rud}) to be valid. \nIn what follows we denote by $\\Psi_-^{\\hbox{\\bf s}}(i,j)$ a polynomial that appears from\n$\\Psi ^{\\hbox{\\bf s}}(i,j)$ given in (\\ref{cbr1}) under the substitutions $x_t\\leftarrow x_t^-,$\n $1\\leq t\\leq n$ with skew commutators defined by (\\ref{sqo}) in $U_q^-(\\frak{ sl}_{n+1}).$ By\npr$W^{\\hbox{\\bf s}}(k,m)$ (respectively, pr$W_-^{\\hbox{\\bf s}}(i,j)$)\nwe denote a subspace spanned by proper derivatives \nof $\\Psi ^{\\hbox{\\bf s}}(k,m)$ (respectively, of $\\Psi_-^{\\hbox{\\bf s}}(i,j)$), see Theorem \\ref{26}. \nConsider two elements \n$\\Psi ^{\\hbox{\\bf s}}(k,m)$ and $\\Psi _-^{T}(i,j).$ Let us display them graphically\nas defined in (\\ref{gra}):\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots \\ & \\stackrel{i-1}{\\bullet } \n\\ & \\stackrel{i}{\\bullet }\\ \\ & \\stackrel{i+1}{\\circ }\\ & \\cdots &\n\\ & \\stackrel{m}{\\bullet } \\ & \\ & \\stackrel{j}{\\cdot } \\cr\n\\ \\ & \\ \\ & \\circ \n\\ & \\circ \\ \\ & \\bullet \\ & \\cdots &\n\\ & \\bullet \\ & \\cdots \\ & \\bullet\n\\end{matrix}\n\\label{gra1}\n\\end{equation}\nWe shall prove that \n\\begin{equation}\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]\\in \n{\\rm pr}\\, W_-^{T}(i,j)\\cdot {\\rm pr}\\, W^{\\hbox{\\bf s}}(k,m)\n\\label{rdi}\n\\end{equation}\nif one of the following two options fulfills:\n\na) Representation (\\ref{gra1}) has no fragments of the form \\label{use}\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{t}{\\circ } \\ & \\cdots & \\stackrel{l}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet \n\\end{matrix}\n\\label{gra2}\n\\end{equation}\n\nb) Representation (\\ref{gra1}) has the form \\label{use1}\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots & \\circ & \\cdots & \\bullet & \\cdots & \\stackrel{m}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet & \\cdots & \\circ & \\cdots & \\bullet \n\\end{matrix}\n\\label{gra3}\n\\end{equation}\n(in particular $i=k,$ $j=m$),\nwhere no one intermediate column has points of the same color.\n\nSuppose that diagram (\\ref{gra1}) satisfies condition a). In this case all black-black\ncolumns are located before all white-white columns. Let us choose the closest\nblack-black and white-white pair of columns. Then (\\ref{gra1}) takes up the form\n\\begin{equation}\n\\underbrace{\n\\begin{matrix}\\ & \\ &\\circ & \\bullet \n& \\bullet & \\cdots \n& \\stackrel{t}{\\bullet } \\cr \n\\cdots & \\circ &\n\\bullet & \\bullet \n& \\circ & \\cdots \n& \\bullet \n\\end{matrix}}_{\\hbox{mainly black}}\\,\n\\underbrace{\n\\begin{matrix} \\stackrel{t+1}{\\circ } & \\bullet & \\cdots \\cr\n\\bullet & \\circ & \\cdots \n\\end{matrix}}_{\\hbox{equality}}\\, \n\\underbrace{\n\\begin{matrix} \\stackrel{l}{\\circ } & \\circ & \\bullet \n& \\circ & \\bullet & \\cdots \\cr\n\\circ & \\circ & \\circ \n& \\bullet & \\\n\\end{matrix}}_{\\hbox{mainly white}}\n\\label{gra4}\n\\end{equation}\nHere in the ``mainly black\" zone there are no white-white columns; in the ``equality\" zone\nwe have just black-white, and white-black columns; while in the ``mainly white\"\nzone there are no black-black columns. Of course the ``mainly black\" zone may be empty.\nIn this case we may omit the ``equality\" zone as well, since all the diagram has no \nblack-black columns at all. In the same way the ``mainly white\" zone may be empty too.\n\nRecall that in Definition \\ref{eski} for a fixed pair $(k,m)$ \nwe define ${\\bf S}_{\\circ }=\\, (${\\bf S}$\\, \\cap [k,m-1])\\, \\cup \\, \\{ k-1 \\} ,$\nwhile ${\\bf S}^{\\bullet }=\\, (${\\bf S}$\\, \\cap [k,m-1])\\, \\cup \\, \\{ m \\} ;$\nrespectively $s_0=k-1\\in \\, {\\bf S}_{\\circ },$ and \n$s_{r+1}=m\\in \\, {\\bf S}^{\\bullet }.$ \n\nAll black-black columns are labeled by numbers from ${\\bf S}^{\\bullet }\\cap T^{\\bullet},$\nwhere the bullets correspond to the pairs $(k,m)$ and $(i,j),$ respectively. \nSimilarly all white-white columns are labeled by numbers from\n $(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ },$\nwhere $\\overline{\\bf S}, \\overline{T}$ are the complements of ${\\bf S},T$ with respect to $[k, m-1],$\n$[i,j-1],$ respectively.\nThus condition a) is equivalent to the inequality \n\\begin{equation}\n\\sup \\{ {\\bf S}^{\\bullet }\\cap T^{\\bullet}\\} <\\inf \\{ (\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }\\} .\n\\label{ineq}\n\\end{equation}\nWe are reminded that the supremum and infimum of the empty set equal $-\\infty $\nand $\\infty ,$ respectively.\n\nCondition b), in turn, means that $i=k,$ $j=m,$ $T=\\overline{\\bf S}.$\n\n\\smallskip\nWe go ahead with a number of useful notes. If $u$ is a word in $X,$ then by $u^-$ we denote\na word in $X^-$ that appears from $u$ under the substitution $x_i\\leftarrow x_i^-.$\nWe have $p(v,w^-)=\\chi ^v(f_w)=p(w,v),$ while $p(w^-,v)=(\\chi ^w)^{-1}(g_v)=p(w,v)^{-1}.$\nThus $p(v,w^-)p(w^-,v)=1.$ Therefore the Jacobi and antisymmetry identities take up their\noriginal ``colored\" form (see, (\\ref{jak1})):\n\\begin{equation} \n[[u,v],w^-]=[u,[v,w^-]]+p_{wv}[[u,w^-],v];\n\\label{uno}\n\\end{equation}\n\\begin{equation} \n[u,w^-]=-p_{wu}[w^-,u].\n\\label{dos}\n\\end{equation}\nIn the same way\n\\begin{equation} \n[[u^-,v^-],w]=[u^-,[v^-,w]]+p_{vw}^{-1}[[u^-,w],v^-].\n\\label{tres}\n\\end{equation}\nUsing antisymmetry and (\\ref{tres}) we have also\n\\begin{equation} \n[u,[v^-,w^-]]=[[u,v^-],w^-]+p_{vu}[v^-[u,w^-]].\n\\label{cua}\n\\end{equation}\nIn these relations $u,v,w$ are words in $X.$ To simplify further\ncalculations we may extend the brackets to the set of all \n$H$-words: We put $\\chi ^{hu}=\\chi ^{u},$ $g_{hu}=hg_u,$\n$h\\in H,$ and define the skew-brackets by the same formula (\\ref{sqo}).\nIn this case we have\n\\begin{equation} \n[u, hv]=\\chi ^u(h)\\, h[u,v], \\ \\ \\ h\\in H;\n\\label{cuq1}\n\\end{equation}\n\\begin{equation} \n[hu,v]=h[u,v]+p_{uv}(1-\\chi^v(h))\\, h\\, v\\cdot u, \\ \\ \\ h\\in H.\n\\label{cuq2}\n\\end{equation}\n\\begin{equation} \n[hu,v]=\\chi^v(h)\\, h[u,v]+(1-\\chi^v(h))\\, h\\, u\\cdot v, \\ \\ \\ h\\in H.\n\\label{cuq21}\n\\end{equation}\nTo calculate the coefficients it is convienient to have in mind the following \nconsequences of (\\ref{a1rel}):\n\\begin{equation} \n\\chi ^k(g_{k-1}f_{k-1})=\\chi ^{k-1}(g_{k}f_{k})=q^{-1}, \\ \\ \\chi^k(g_if_i)=1,\\ \\hbox{ if } |i-k|>1.\n\\label{cuq22}\n\\end{equation}\nOf course all basic formulae (\\ref{jak1}), (\\ref{jak3}), (\\ref{br1}) and their consequences \nremain valid. However we must stress that\nonce we apply relations (\\ref{rela3}), or other ``inhomogeneous in $H$\" relations\n(for example the third option of (\\ref{derm1}), see below), \nwe have to fix the curvature of the brackets as soon as\nthe inhomogeneous substitution applies to the right factor in the brackets:\n\\begin{equation}\n[u,[x_i,x_i^-]]=u(1-g_if_i)-\\chi ^u(g_if_i)(1-g_if_i)u=(1-\\chi^u(g_if_i))u,\n\\label{cuq3}\n\\end{equation}\nbut not $[u,[x_i,x_i^-]]=[u,1-g_if_i]=[u,1]-[u,g_if_i]=0.$ At the same time\n\\begin{equation}\n[[x_i,x_i^-],u]=(1-g_if_i)u-u(1-g_if_i)=(\\chi^u(g_if_i)-1)\\, g_if_i\\cdot u,\n\\label{cuq4}\n\\end{equation}\nand $[[x_i,x_i^-],u]=[1-g_if_i, u]$ $=[1,u]-[g_if_i,u]$ is valid since the inhomogeneous\nsubstitution has been applied to the left factor in the brackets. In what follows we shall \ndenote for short $h_i=g_if_i,$ and $\\bar{h}_{ki}=h_kh_{k+1}\\cdots h_{i-1},$\nwhere $kk. \\hfill \\end{matrix} \\right.\n\\label{derm2}\n\\end{equation}\nLet us put $u=\\Psi ^{\\hbox{\\bf s}}(1+s_1,m),$ $v=u[k,s_1],$ $w^-=x_k^-.$\nThe definition (\\ref{cbr1}) shows that $\\Psi ^{\\hbox{\\bf s}}(k,m)=[u,v],$\nwhile (\\ref{uno}) implies $[[u,v],w^-]=[u,[v,w^-]].$\n\n If $s_1=k,$ then by (\\ref{cuq3}) we have $[u,[v,w^-]]$\n$=u-\\chi ^u(h_k)u=(1-q^{-1})u.$\n\nIf $s_1>k,$ then (\\ref{derm1}) yields\n$[v,w^-]\\sim h_k\\cdot u[k+1,s_1].$ Therefore $[u,[v,w^-]]\\sim h_k[u, u[k+1,s_1]],$\nsee (\\ref{cuq1}).\nIt remains to note that $[u, u[k+1,s_1]]=\\Psi ^{\\hbox{\\bf s}}(k+1,m)$ due to (\\ref{cbr1}).\n Thus formula (\\ref{derm2}) is proved.\n\\begin{equation}\n[\\Psi ^{\\hbox{\\bf s}}(k,m), x_m^-]\\sim \n\\left\\{ \\begin{matrix}h_m\\cdot \\Psi ^{\\hbox{\\bf s}}(k,m-1),\\hfill & \\hbox{if }k\\leq s_r=m-1;\\hfill \\cr\n\\Psi ^{\\hbox{\\bf s}}(k,m-1), \\hfill & \\hbox{if } k\\leq s_rm.$ In this case $\\mu =m.$ We use induction on $m-k.$ If $m$ $=k,$\nthe formula follows from dual (\\ref{derm1}). If $m>k,$ we put\n$u=\\Psi ^{\\hbox{\\bf s}}(1+s_1,m),$\n$v=u[k,s_1],$\n$w^-=u[k,j]^-.$\nAccording to (\\ref{cbr1}) we have $[u,v]$ $=\\Psi ^{\\hbox{\\bf s}}(k,m),$\nwhile $[u,w^-]=0$ due to (\\ref{rdi1}) with $T=\\emptyset ,$ $T_{\\circ }=\\{ a-1\\} .$\nBy Jacobi identity (\\ref{uno}) and (\\ref{dm3})\nwe get\n\\begin{equation}\n[\\Psi ^{\\hbox{\\bf s}}(k,m),w^-]=[u,[v,w^-]]\n=\\hbox{\\huge [}u, \\sum _{a=k+1}^{1+s_1}\\alpha _a\\, \\bar{h}_{ka}\\, \nu[a,j]^-\\cdot u[a,s_1]\\hbox{\\huge ]}.\n\\label{rrr}\n\\end{equation}\nRelation (\\ref{rdi1}) with $T=\\emptyset ,$ $T_{\\circ }=\\{ k-1\\} $ implies\n$[u,u[a,j]^-]$ $=0$ unless $a=1+s_1.$ Using ad-identities (\\ref{br1}), (\\ref{cuq1})\nwe may continue\n$$\n=\\alpha \\, \\bar{h}_{k\\, 1+s_1}\\, [u,u[1+s_1,j]^-]\n+\\sum _{a =k+1}^{s_1}\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n$$\nwhere $\\alpha \\neq 0.$\nSince $1+s_1>k,$ we may apply the inductive supposition to the first summand.\n\nLet $jm$ with $m\\leftarrow s_l$ we get\n$$\n[\\Psi ^{\\hbox{\\bf s}}(k,m),w^-]=[u,[v,w^-]]=\n\\hbox{\\large [}u,\\sum _{a=k+1}^{1+s_l}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,s_l) \\hbox{\\large ]},\n$$\nwhere $\\alpha _a=0$ if $a-1\\in \\, \\hbox{\\bf S}\\, \\cap \\, [k,s_l-1].$\n Relation (\\ref{dm1}) imply\n$[u,u[a,j]^-]=0$ unless $a=1+s_l.$\nHence by ad-identities (\\ref{br1}), (\\ref{cuq1}) we may continue\n$$\n=\\alpha \\, \\bar{h}_{k\\, 1+s_l}\\, [u,u[1+s_l,j]]+\n\\sum _{a=k+1}^{s_l}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n$$\nwhere $\\alpha \\neq 0,$ $\\alpha _a=0$ if $a-1\\in \\, \\hbox{\\bf S}\\, \\cap \\, [k,s_l-1].$\nIt remains to apply (\\ref{dm3}) to the first summand.\n\nIf, finally, $l\\leq r,$ we put\n$u=\\Psi ^{\\hbox{\\bf s}}(1+s_{l+1},m),$\n$v=\\Psi ^{\\hbox{\\bf s}}(k,s_{l+1}),$\n$w^-=u[k,j]^-.$\nThen still $[u,v]$ $=\\Psi ^{\\hbox{\\bf s}}(k,m),$ see decomposition (\\ref{cbrr}),\nand $[u,w^-]=0.$ By the considered above case with $m\\leftarrow s_{l+1}$ we have\n$$\n[v,w^-]=\n\\sum _{a=k+1}^{j+1}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,s_{l+1}),\n$$\nwhere $\\alpha _a=0$ if $a-1\\in \\, {\\bf S}\\, \\cap [k,s_{l+1}-1].$\nIn this case $[u,u[a,j]^-]=0$ since $ji=k.$ If $T=\\emptyset ,$ the formula is already proved, see (\\ref{dm31}). \nSuppose that $T\\neq \\emptyset .$\nLet us denote $u=\\Psi ^{\\hbox{\\bf s}}(k,m),$ $v^-=\\Psi ^{T}_-(1+t_1,j),$\n$w^-=u[k,t_1].$ Then according to definition (\\ref{cbr1}) the left hand side of (\\ref{rdi2})\nequals $[u,[v^-,w^-]].$ By (\\ref{rdi1}) we have $[u,v^-]=0.$ Hence Jacobi identity\n(\\ref{cua}) implies $[u,[v^-,w^-]]\\sim [v^-,[u,w^-]].$ Using (\\ref{dm31}) we get\n\\begin{equation}\n[u, w^-]=\\sum _{a=k+1}^{\\mu +1}\\alpha _a \\,\n\\bar{h}_{ka} \\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n\\label{dmr}\n\\end{equation}\nwhere $\\mu ={\\rm min }\\{ m, t_1\\},$ while $\\alpha _a=0$ if \n$a-1\\in \\, ${\\bf S}$\\, \\cap \\, [k,m-1],$ and formally \n$\\Psi ^{\\hbox{\\bf s}}(m+1,m)=[j+1,j]^-=1.$ \nOf course, $m\\neq t_1$ since $S^{\\bullet }\\cap T^{\\bullet }=\\emptyset .$\n\nIf $m>t_1,$ then we have\n$$\n[v^-,[u, w^-]=\\alpha _{1+t_1} \\,\n[v^-, \\bar{h}_{k\\, 1+t_1} \\, \\Psi ^{\\hbox{\\bf s}}(1+t_1,m)]+\n\\sum _{a=k+1}^{t_1}\\alpha _a \\,\n[ v^-,\\bar{h}_{ka} \\, u[a,t_1]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m)].\n$$\nBy (\\ref{dos}) and (\\ref{rdi1}) we have $[v^-, \\Psi ^{\\hbox{\\bf s}}(a,m)]=0$ if $kk.$ In this case $k-1\\in T\\cap [i,j-1],$ say $k=1+t_l.$\nLet us put $u=\\Psi ^{\\hbox{\\bf s}}(k,m),$ $v^-=\\Psi ^T_-(k,j),$ $w^-=\\Psi ^T_-(i,k-1).$\nDecomposition (\\ref{cbrr}) implies $\\Psi ^T_-(i,j)=[v^-,w^-].$ Since $[u,w^-]=0,$\nwe have $[u,[v^-,w^-]]=[[u,v^-],w^-].$ To find $[u,v^-]$ we may use already considered case: \n$$\n[u,v^-]=\\sum _{a=k+1}^{\\mu +1}\\alpha _a\\bar{h}_{ka}\\Psi ^T_-(a,j)\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m)\n$$\nwith $\\alpha _a=0$ if $a-1\\in {\\bf S}\\cup T,$ $a\\neq \\mu +1.$ Certainly\n$[\\Psi ^{\\hbox{\\bf s}}(a,m), w^-]$ $=[\\Psi ^T_-(a,j), w^-]=0$ since $a>k.$\nBy means of (\\ref{cuq22}) we have\n$$\n\\chi ^{w^-}(\\bar{h}_{ka})=\\chi ^{w}_-(h_kk_{k+1}\\cdots h_{a-1})=\\chi ^{k-1}_-(h_k)=q\\neq 1.\n$$\nNow formula (\\ref{cuq2}) shows that $[[u,v^-],w^-]\\sim w^-\\cdot [u,v^-],$ which is required.\n\n\nIn perfect analogy, if $ki.$\nBy means of (\\ref{cuq22}) we have\n$$\n\\chi ^{v}(\\bar{h}_{ia})=\\chi ^{v}(h_ih_{i+1}\\cdots h_{a-1})=\\chi ^{i-1}(h_i)=q^{-1}\\neq 1.\n$$\nNow formula (\\ref{cuq21}) shows that $[[u,w^-],v]\\sim [u,w^-]\\cdot v,$ which is required.\n\\end{proof}\n\nWe have mentioned above that our main concepts (Definition \\ref{eski}) are not invariant\nwith respect to the replacement of $x_i,$ $x_i^-,$ $1\\leq i\\leq n$\nby $y_i,$ $y_i^-,$ $1\\leq i\\leq n,$ where by definition $y_i=x_{\\varphi (i)},$\n$y_i^-=x_{\\varphi (i)}^-,$ $\\varphi (i)=n-i+1.$\nHence the application of already proved lemmas to generators $y_i,$ $y_i^{-}$\nprovides an additional information. In this way we are going to prove the following two statements.\n\n\\begin{lemma} \nIf $(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }\n=(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }=\\emptyset $ then\n\\begin{equation}\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]=0.\n\\label{rdi1b}\n\\end{equation}\n\\label{zerb}\n\\end{lemma}\n\n\\begin{lemma} \nIf \n$(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }= \\emptyset ,$ while\n$(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }\\neq \\emptyset $\n then\n$$\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]\n$$\n\\begin{equation}\n=\\Psi ^T_-(\\mu +1,j)\\left(\n\\sum _{b=\\nu -1}^{\\mu -1}\\alpha _{b}\\, \n\\bar{h}_{b+1 \\, \\mu +1}\\, \\Psi ^T_-(i,b)\\cdot \\Psi ^{\\hbox{\\bf s}}(k,b)\\right) \\Psi ^{\\hbox{\\bf s}}(\\mu +1,m)\n\\label{rdi2b}\n\\end{equation}\nwhere $\\mu =\\min \\{ m,j\\} ,$ $\\nu =\\max \\{ k,i \\} ,$\nwhile $\\alpha _{b}=0$ provided that $b\\notin {\\bf S} \\cap T$\nwith the only exception, $\\alpha _{\\nu -1}\\neq 0.$\n\\label{zerb1}\n\\end{lemma}\n\\begin{proof}\nTo derive these statements from Lemma \\ref{zer} and Lemma \\ref{zer1}\nwe use the {\\it decoding} lemma, Lemma \\ref{dec}.\nLet us apply (\\ref{decod}) to the left hand side of (\\ref{rdi2b}). In this case we have\n$$\n({\\overline{\\varphi(\\hbox{\\bf S})-1}})_{\\circ }=\\varphi \\{ (\\overline{\\bf S})^{\\bullet }+1\\},\\ \n({\\overline{\\varphi(\\hbox{\\bf S})-1}})^{\\bullet }=\\varphi \\{ (\\overline{\\bf S})_{\\circ }+1\\},\n$$\nwhere in the left hand sides the operators (Definition \\ref{eski}) correspond\nto $(\\varphi (m), \\varphi (k)), $ while in the right hand sides to $(k,m).$ In particular \n$(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }=\\emptyset $ is equivalent to\n$({\\overline{\\varphi (\\hbox{\\bf S})-1}})_{\\circ }\\cap ({\\overline{\\varphi (T)-1}})_{\\circ } =\\emptyset ,$\nwhile\n$(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }=\\emptyset $ is equivalent to \n$({\\overline{\\varphi (\\hbox{\\bf S})-1}})^{\\bullet }\\cap ({\\overline{\\varphi (T)-1}})^{\\bullet } =\\emptyset .$\nHence we may \nuse relations (\\ref{rdi1}), (\\ref{rdi2}). Relation (\\ref{rdi1}) proves Lemma \\ref{zerb}.\nIn the case of Lemma \\ref{zerb1} we \nagain apply decoding formula (\\ref{decod}) in order to get (\\ref{rdi2b}). \\end{proof}\n\n\\begin{proposition} \nCondition $a),$ p.$\\pageref{use}$ implies $(\\ref{rdi}).$\n\\label{coa}\n\\end{proposition}\n\\begin{proof} \nWe have seen that condition a) is equivalent to inequality (\\ref{ineq}).\n\nIf ${\\bf S}^{\\bullet }\\cap {T}^{\\bullet }=\\emptyset $ \nthen we may use Lemma \\ref{zer} and Lemma \\ref{zer1}. Let us show that all factors in\n(\\ref{rdi2}) in terms with nonzero $\\alpha _a$ belong to either pr$\\, W^{T}_-(i,j)$\nor pr$\\, W^{\\hbox{\\bf s}}(k,m).$ \n\nIf $a=\\mu +1,$ say $a-1=m2,$ then this is the case for $\\Gamma $-homogeneous\nright coideal subalgebras of $u_q(\\frak{ sl}_{n+1}).$\n\\label{osn5}\n\\end{theorem}\n\\begin{proof} By Lemma \\ref{raz1} we have to show that \n$[U_{\\theta }^+,U_{\\theta ^{\\prime }}^-]\\subseteq {\\bf U}.$\nThe first condition in the theorem means that $\\Psi ^{T_k}(k,\\tilde{\\theta }_k)$ and\n$\\Psi_-^{T_i^{\\prime }}(k,\\tilde{\\theta }_i^{\\prime })$ satisfy condition a) p.\\pageref{use},\nwhile the second one is equivalent to condition b) p.\\pageref{use1} for these elements.\nBy definition $[k:\\tilde{\\theta }_k]$ is a simple $U_{\\theta }^+$-root, such that any other \nsimple $U_{\\theta }^+$-root of the form $[k:m]$ satisfies $m<\\tilde{\\theta }_k.$\nSimilarly each simple $U_{\\theta ^{\\prime }}^-$-root of the form $[i:j]$\nsatisfies $j\\leq \\tilde{\\theta }_i^{\\prime }.$ Condition a) certainly remain valid\nfor subdiagrams, while proper subdiagrams of (\\ref{gra3}) satisfy condition a).\nTherefore, due to Proposition \\ref{coa} and Proposition \\ref{cob}, for each pair of a simple \n$U_{\\theta }^+$-root, $[k:m],$ and a simple $U_{\\theta ^{\\prime }}^-$-root, $[i:j],$ we have \n\\begin{equation}\n [\\Psi ^{T_k}(k,m),\\Psi _-^{T_i^{\\prime }}(i,j)]\\in {\\bf U}.\n\\label{rdii}\n\\end{equation}\nBy Claim 8 the algebras $U^+_{\\theta },$ and $U^-_{\\theta ^{\\prime }}$\n are generated by $\\Psi ^{T_k}(k,m),$\nand $\\Psi _-^{T_i^{\\prime }}(i,j),$ respectively, where $[k:m]$ and $[i:j]$ run through \nthe sets of simple roots. \nTo show that $[U^+_{\\theta },U^-_{\\theta ^{\\prime }}]\\subseteq {\\bf U},$\nit remains to apply ad-identities (\\ref{br1f}), (\\ref{br1}) and evident induction\non degree (we remark that in (\\ref{rdi}) the degree of factors diminishes).\n\nConversely, suppose that $[U^+_{\\theta },U^-_{\\theta ^{\\prime }}]\\subseteq {\\bf U}.$\nLet us choose any pair $(k,i),$ and denote \n$$\nt=\\sup \\left\\{ a\\, |\\, k\\leq a\\leq \\tilde{\\theta }_k,\\, i\\leq a\\leq \\tilde{\\theta }_i^{\\prime },\n\\, a\\in T_k, \\, a\\in T_i^{\\prime } \\right\\},\n$$\n$$ \nl= \\inf \\left\\{ b\\, |\\, k-1\\leq b< \\tilde{\\theta }_k,\\, i-1\\leq b< \\tilde{\\theta }_i^{\\prime },\n\\, b\\notin T_k, \\, b\\notin T_i^{\\prime } \\right\\}.\n$$\n If one of these sets is empty then \ncondition (\\ref{yo1}) is valid. Suppose that $t 0 \\\\\n\\rm & 0 \t\t& \t\\ \\hat{\\sigma_p} < 0\n\\end{array}\\right.\n\\end{equation}\nwith, \n\\begin{equation}\n\\lambda(0) = \\frac{\\mathscr{L}(\\sigma_p = 0,\\hat{\\hat{R_b}})}{\\mathscr{L}(\\hat{\\sigma_p},\\hat{R_b})}\n\\end{equation}\nHence, a large value of $q_0$ implies a large discrepancy between \nthe two hypothesis which is in favor of a discovery ($H_1$). As $f(q_0 \\mid H_0)$ follows a $\\chi^2_1$ distribution, the discovery significance $Z$ \nis simply defined as $Z = \\sqrt{q^{\\rm obs}}$, in units of $\\sigma$ \\cite{Cowan:2010js}.\\\\ \n\n\n\nThe second approach, first introduced by B. Morgan {\\it et al.} \\cite{morgan2}, is based on a generic test of isotropy following the mean recoil deviation $\\langle \\cos\\theta\n\\rangle$ such as:\n\\begin{equation}\n\\langle \\cos{\\theta} \\rangle = \\frac{1}{N}\\sum^N_{i=1}{\\cos{\\theta_i}}\n\\end{equation}\nwhere $\\theta_i$ is the $i^{th}$ angle between the recoil and the Cygnus direction, and $N$ is the number of\nmeasured recoils. Note that this test, as well as the previous one, is by definition coordinate system dependent as the main recoil direction\n$(\\ell,b)$ (see \\cite{billard.disco}) is not considered here as a fitting parameter.\\\\\nEventually, one can evaluate the significance of an observed anisotropy by computing the distributions of $\\langle\\cos\\theta\\rangle$ for\nboth $H_0$ corresponding to the background (isotropic) and $H_1$ the alternative. It is worth noticing that the use of the \n variable $\\langle\\cos\\theta\\rangle$ is particularly interesting in the case of directional detection of Dark Matter as the expected signal should exhibit a dipole feature\n hence maximizing the deviation between $H_0$ and $H_1$.\n\n\n\n\n\n\\section{Influence of a co-rotating Dark Disk}\n\\label{sec:disco}\n \n\nIn order to investigate the effect of a Dark Disk component on the expected significance of a directional dark matter detection, \nwe allow for a wide range on the Dark Disk parameters, see eq.~\\ref{eq:ddparam}, and we evaluate, for each configuration, \nthe expected significance for a 30 kg.year MIMAC-like detector. We highlight the fact that for a co-rotating Dark Disk to contribute to the data, \nthe energy threshold must be low and\/or the WIMP mass large. For concreteness, we present a case study for a 50 GeV\/c$^2$ WIMP \nmass and a total of 100 WIMP events. Figure~\\ref{fig:dispersion} (left) presents the mean significance E(Z) as a function of $V_{DD}$, the rotation velocity of the Dark Disk at \nSolar radius. The black dashed line corresponds to the no Dark Disk case. The result is then presented for various values \nof the relative density $\\rho_{DD}\/\\rho_{H}$. The general feature is that the mean significance is\ndecreasing when increasing the co-rotating velocity of the Dark Disk at Solar radius as it results in a loss of \ndirectionality. This effect is even stronger when increasing the Dark Disk contribution, {\\it i.e.} for large values of the relative velocity $\\rho_{DD}\/\\rho_{H}$. \nInterestingly, a co-rotating Dark Disk can boost the mean significance of a Directional Dark Matter detection. Indeed, for a\nvelocity dispersion $\\sigma_{DD} = 85$ km\/s and a WIMP mass of 50 GeV\/c$^2$, one can see that for rotation velocity $V_{DD} \\leq 140$ \nkm\/s and for any relative density, the mean significance obtained is greater than the one obtained in the no Dark Disk case.\nThis enhancement of the significance at low rotation velocities can be explained by the fact that the Dark Disk is a structure colder than the host halo, {\\it i.e.} has a smaller \nvelocity dispersion, implying an even more anisotropic recoil angular distribution. Hence, a co-rotating Dark Disk will not\nnecessarily degrade the expected performance of directional detection. Of course, for a perfectly \nco-rotating Dark Disk ($V_{DD} = v_\\odot = 220$ km\/s), whatever the velocity dispersion, the recoil angular distribution induced by the \nDark Disk is necessarily isotropic. Only the contribution of the Dark Disk to the total number of events will change.\\\\\nFigure~\\ref{fig:dispersion} (right) presents the mean significance as a function of $V_{DD}$. For any value of the \nvelocity dispersion $\\sigma_{DD}$, the mean significance is continuously \ndecreasing with the rotation velocity of the Dark Disk. However, the case $\\sigma_{DD} = 35$ km\/s (red solid line) \ntends to the no Dark Disk limit due to the\nfact that the contribution to the total number of WIMP events from the Dark Disk falls quickly to zero for $V_{DD} > 140$ km\/s. \nThis also explains the rapid decrease of the significance enhancement in the range $0 - 100$ km\/s. For larger velocity dispersions, \nthe mean significance does not tend to the no Dark Disk limit as the contribution of the Dark Disk to the total number of events \nremains non negligible. Interestingly, one may note that the range of the values of\n $V_{DD}$ inducing an enhancement of the significance depends strongly on $\\sigma_{DD}$. Indeed, \n for large values of the \n velocity dispersion, the Dark Matter signal gets closer to an isotropic distribution. \n This observation implies that lower is the velocity\n dispersion, larger is the range in values of $V_{DD}$ allowing for a boost of the directional signature. \n As a conclusion, larger is the velocity dispersion of the Dark Disk, weaker is the directional discovery significance, \n except for the case of $\\sigma_{DD} = 35$ km\/s as discussed above. However, note that the value of $\\sigma_{DD} = 141$ km\/s\n is extremely large with respect to the recent results from N-Body simulations. Hence, for a 50 GeV\/c$^2$ WIMP mass, one could expect that a co-rotating Dark Disk could \n have a positive, though small ($\\sim$ 10\\%), effect on the directional detection of Dark Matter.\n \n\n\n\n\n\nFor completeness, we studied the evolution of the modifications of the angular distribution $dR\/d\\Omega_r$ for various Dark Disk parameter values. For this purpose, we\ndefined the relative asymmetry $\\mathscr{A}$ as:\n\\begin{equation}\n\\mathscr{A} = \\frac{\\langle\\cos\\theta\\rangle - \\langle\\cos\\theta\\rangle_{H}}{\\langle\\cos\\theta\\rangle_{H}}\n\\label{eq:a}\n\\end{equation}\nwhere $\\langle\\cos\\theta\\rangle_{H}$ corresponds to the mean recoil deviation obtained in the no Dark Disk case ($\\rho_{DD} = 0$). Note that considering the standard halo\nmodel and a WIMP of 50 GeV\/c$^2$, we found $\\langle\\cos\\theta\\rangle_{H} \\simeq 0.51$. Figure~\\ref{fig:dispersion2} presents the relative asymmetry $\\mathscr{A}$ \nin the plane ($V_{DD}, \\sigma_v^{DD}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (left) and $\\rho_{DD}\/\\rho_H = 1$ (right). One may notice that there are three\ndifferent regions : no effect (the 1\\% region), a directional discovery enhancement region (low $V_{DD}$) and a region for which the Dark Disk weakens the directional\nsignature (high $V_{DD}$ together with a high $\\sigma_{DD}$ value). The relative density only affects the amplitude of $\\mathscr{A}$, note that the latter spans the range\n[-18,10] for $\\rho_{DD}\/\\rho_H = 1\/3$ and [-40,23] for $\\rho_{DD}\/\\rho_H = 1$. This also affects the area of the no effect region which \n decreases with increasing value of $\\rho_{DD}\/\\rho_{H}$. Interestingly, one can notice that most of the Dark Disk models suggested by N-Body simulations lie in the no effect\n region. Only extreme, yet unrealistic, Dark Disk models may affect significantly the directional signature. It corresponds to the case when both the co-rotational velocity and the velocity dispersion are\n high.\\\\\n\nThis result is in good agreement with previous work \\cite{Green:2010gw} on the effect of a Dark Disk component on directional detection reach which led to the following conclusion. \nThere is only a small variation, with respect to the no Dark Disk case, in the number of WIMP events required to reject isotropy (at 95\\% confidence in 95\\% of experiments) \nor to reject the median direction being random (at 95\\% confidence in 95\\% of experiments). \nNote that the study has been done for a Sulfure detector (a DRIFT-like one) assuming a 20 keV energy threshold and no background.\\\\\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{Vdd_Sigdd_Rho0333.eps}\n\\includegraphics[scale=0.5,angle=0]{Vdd_Sigdd_Rho1.eps}\n\\caption{ Relative asymmetry $\\mathscr{A}$ in the plane ($V_{DD}, \\sigma_{DD}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom).\nThe solid and dashed lines correspond to isocontours of a relative asymmetry equal to $\\pm 1$\\% and $\\pm 5$\\% respectively. \nThese studies have been done for 50 GeV\/c$^2$ WIMP mass.} \n\\label{fig:dispersion2}\n\\end{center}\n\\end{figure*} \n\nSo far, we have focused on an isotropic Maxwellian distribution for the Dark Disk particles. \nHowever, as shown in \\cite{Ling:2009cn}, the dark disk itself may exhibit\nanisotropic features in its velocity distribution. One way to investigate its\nvelocity dispersion tensor using current experimental data, is to look at the stellar thick disk. However, comparison with the observed values of the velocity dispersions of the stellar thick disk may be misleading as the Dark Disk \nanisotropy depends strongly on the merger properties such as infall inclinations. Nevertheless, evidence in favor of a departure from \nisotropy comes from full cosmological hydrodynamics simulations \\cite{Ling:2009cn}.\\\\ \nTo study the effect of an anisotropic Dark Disk, \nwe evaluate the value of the relative asymmetry $\\mathscr{A}$ (eq.~\\ref{eq:a}) as a function of the radial dispersion $\\sigma_r$ and \nthe tangential one, defined as $\\sigma_t^2=\\sigma_y^2+\\sigma_z^2$. Figure \\ref{fig:dispersion3} presents the \nrelative asymmetry $\\mathscr{A}$ in the plane ($\\sigma_{r}, \\sigma_{t}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom). These studies have been done for a \nWIMP mass of 50 GeV\/c$^2$ and a co-rotational velocity $V_{DD} = 150 \\ km\/s$. \nFor convenience the isovalues of the anisotropy parameter $\\beta = 1 - \\sigma_t^2\/2\\sigma_r^2$ are indicated. \nNote that a positive and a negative value of $\\beta$ refer to a radially and tangentially \nanisotropic velocity distribution respectively.\nFirst, it can be noticed that, for a fixed value \nof $\\sigma_t$, the relative asymmetry decreases with increasing $\\sigma_r$, \n{\\it i.e.} perpendicularly to the detector motion direction, as the WIMP flux is becoming more isotropic \nin the detector frame, without enhancing the Dark Disk contribution to the data (see discussion above). \nFor a fixed value of $\\sigma_r$, due to the Earth rotation along the $(Oy)$ axis, a larger dispersion along this axis \nwill mostly boost the Dark Disk contribution to the number of WIMPs events while keeping a strong anisotropy, if $\\sigma_t$ is \nnot too large. Hence, there is an optimal point above which, increasing $\\sigma_t$ and hence both $\\sigma_y$ and $\\sigma_z$ starts to make the flux sufficiently anisotropic to weaken the directional signal.\nEventually it should be highlighted that for any departure from isotropy, the effect on the relative asymmetry remains small, -15\\% at the very\nmost and in the extreme cas of a relative density of $\\rho_{DD}\/\\rho_H = 1$. On its own, the effect of the Dark Disk anisotropy is small compared to the influence coming from the standard parameters such as the one previously studied ($\\sigma_{DD}$ and $V_{DD}$).\\\\\n\n \n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{SigR_Sig_t_Rho0333_corrected.eps}\n\\includegraphics[scale=0.5,angle=0]{SigR_Sig_t_Rho1_corrected.eps}\n\\caption{Relative asymmetry $\\mathscr{A}$ in the plane ($\\sigma_{r}, \\sigma_{t}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom). The solid and dashed lines correspond to isocontours of a relative asymmetry equal to $\\pm 1$\\% and $\\pm 5$\\% respectively. \nThese studies have been done for a WIMP mass of 50 GeV\/c$^2$ and a co-rotational velocity $V_{DD} = 150 \\ km\/s$.} \n\\label{fig:dispersion3}\n\\end{center}\n\\end{figure*} \n\n\n\nWe evaluate the effect of the Dark Disk contribution to the Dark Matter \nreach of upcoming directional detectors. Following \\cite{billard.profile}, we compute the directional reach in the $(m_{\\chi}, \\sigma_p)$ plane, {\\it i.e} the \nlower bound of the 3$\\sigma$ discovery region at 90\\% CL for the two approaches: profile likelihood (red lines) and mean recoil deviation (blue lines).\n Figure \\ref{fig:ProspectsFinal} presents the \ndiscovery limit in the ($m_\\chi,\\log_{10}(\\sigma_p)$) plane corresponding to two Dark Matter models: standard halo model only (solid lines) and \n and with an extreme Dark Disk model contribution \\{$\\rho_{DD}\/\\rho_h = 1$, $V_{DD} = 220$ km\/s, $\\sigma_{DD} = 106$ km\/s\\} (dashed lines). \\\\\n The conclusion of this study is twofold. First, we found that for both statistical tests, the effect of an extreme Dark Disk is only mild. Indeed, the directional reach is\n only degraded by a factor of 3 at high WIMP masses and not affected for light WIMP. Second, we found that the two statistical tests give similar results with a maximal\n deviation of a few percent. However, it is worth emphasizing that their intepretation differ as the profile likelihood method favors\n the background plus signal hypothesis ($H_1$)\n whereas the mean recoil deviation method rejects the isotropy hypothesis.\n \n\n \n \n \\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{ProspectsDarkDisk.eps}\n\\caption{Discovery limit in the ($m_\\chi,\\log_{10}(\\sigma_p)$) plane corresponding to two Dark Matter models: standard halo model only (solid lines) and \n and with an extreme Dark Disk model contribution \\{$\\rho_{DD}\/\\rho_h = 1$, $V_{DD} = 220$ km\/s, $\\sigma_{DD} = 106$ km\/s\\} (dashed lines). \n We compute the directional reach, {\\it i.e} the \nlower bound of the 3$\\sigma$ discovery region at 90\\% CL, for the two approaches: profile likelihood (red lines) and mean recoil deviation (blue lines).} \n\\label{fig:ProspectsFinal}\n\\end{center}\n\\end{figure}\n\n \n \n\n\n\n\\section{Conclusion}\nA co-rotating Dark Disk, as predicted by recent N-Body simulations, might contribute \n(10\\%-50\\%) to the local Dark Matter density, with a potentially dramatic effect on directional detection. \nIn this letter, we have evaluated the effect of Dark Disk model on the discovery potential \nof upcoming directional detectors. We conclude that, if a co-rotating Dark Disk is present in our Galaxy and has the properties predicted by N-Body simulations \\cite{nezri}, \n the discovery potential of directional detection would be strictly unchanged. Only an extreme and unrealistic Dark Disk model (high co-rotational velocity and high velocity dispersion) \n might affect significantly the Dark Matter reach of upcoming directional detectors, by increasing the discovery limit by a \n factor of three at high WIMP mass ($m_{\\chi} \\sim 1000$ GeV\/c$^2$). Additionally, we also have shown that anisotropic features in the Dark Matter velocity distribution of the Dark Disk will only have a small effect on the expected directional signal. Hence, according to our results we believe that the possibility of the existence of a co-rotational Dark Disk in our galaxy shouldn't be a threat for upcoming directional detection experiments.\\\\\n Interestingly, note that even if the impact of Dark Disk contribution to the local Dark Matter distribution only mildly affects the discovery \n potential of directional detection, it may\n significantly affect the mass and cross section determination \\cite{billard.ident}. Indeed, as explained in \\cite{Green:2010gw}, \n WIMP events arising from the Dark Disk contribution will induce an\n excess at low recoil energies which can lower the estimation of the WIMP mass when considering a standard halo model. \n As outlined in \\cite{Lee:2012pf}, the presence of a Dark Disk restricts the ability to constrain \n the Dark Matter parameters (both from the halo and particle physics). Of course, a measurement of the parameters of the Dark Disk itself remains challenging with the exposure \n of the next generation of directional detectors (30 kg.year). This highlights the fact that even if a co-rotating Dark Disk is not a threat to the discovery potential of directional detection, \n it has to be characterized in order to consistently constrain the Dark Matter properties. \n \n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}