diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfnat" "b/data_all_eng_slimpj/shuffled/split2/finalzzfnat" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfnat" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nStochastic models have been extensively used in theoretical neuroscience since the pioneer work by Gerstein and Mandelbrot in 1964 \\cite{GernstenMandelbrot}. There they considered a Wiener process (also known as Brownian motion or Perfect-Integrate-and-Fire model) to model the voltage across the membrane. An action potential, also known as spike, is generated whenever the membrane potential reaches a certain constant threshold. After that, the membrane voltage is reset to its resting value and the evolution restarts. From a mathematical point of view, a spike is the first passage time (FPT) of a stochastic process to a constant threshold. The collection of spike epochs of a neuron, called spike train, defines a renewal process, with independent and identically distributed inter-spike intervals (ISIs). Despite the excellent fit with some experimental data, the Gerstein-Mandelbrot model was criticized because it disregards features involved in neuronal coding.\n\nA first extension, combining both mathematical tractability and biological realism, is represented by \\emph{Leaky-Integrate-and-Fire} (LIF) models \\cite{ReviewSac,Tuckwell88}. Despite some criticisms on the lack of fit of experimental data \\cite{Jolivetetal,Shinomoto}, these models are still largely used.\n\nAnother common generalization is represented by Wiener processes (or more generally LIF models) with \\emph{time-dependent threshold} \\cite{Tuckwell78,TuckwellWam}. These models can be chosen to reproduce biological features such as the afterhyperpolarization in neurons. For exponentially decaying thresholds, these processes can be used to model a neuron with an exponential time-dependent drift, as shown by Lindner and Longtin \\cite{LindnerLongtin}. They investigated the effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron \\cite{LindnerLongtin}. Using a perturbation method\n\\cite{Lindner2004b}, they derived analytical expressions of the firing statistics under the assumption that the amplitude $\\epsilon$ of the time-dependent change in the threshold is small. These statistics are useful to characterize the spontaneous neural activity and to investigate the neuronal signal transmission. In particular, they can suggest under which conditions a decaying threshold may facilitate or deteriorate signal processing by stochastic neurons. For a Wiener process, these quantities can also be obtained using the approach in \\cite{Urdapilleta}. Also this method assumes a small amplitude $\\epsilon$, but it has the advantage of providing an explicit approximation of the FPT density.\n\nHere we consider a Wiener process with exponentially decaying threshold. The first aim of the paper is to provide an alternative method to approximate the firing statistics and the FPT density for any possible amplitude $\\epsilon$, extending the results in \\cite{LindnerLongtin, Urdapilleta}. Different estimators are proposed, as mentioned in Section \\ref{Sec1a} and discussed in Section \\ref{Sec4b}. Means, variances, coefficients of variation (CVs) and distributions of the FPTs are compared on simulated data and the most suitable are recommended. A comparison with the results in \\cite{LindnerLongtin, Urdapilleta} under the assumption of a small amplitude $\\epsilon$ is also performed. The second aim of this work is the estimation of drift and diffusion coefficients of the Wiener process. Maximum likelihood and moment estimators are derived and evaluated on simulated data. Our results show a good approximation of both firing statistics and parameters of the underlying model.\n\nAlthough the considered model generates a renewal process, the proposed method can also be applied to non-renewal processes, e.g. adaptive threshold models \\cite{Chacron3,Kobayashi1}. Recently, an increasing interest arose towards these models, interest motivated by the excellent fit of the firing statistics of electrosensory neurons \\cite{Chacron2,Chacron1}. The novelty of these models is that the threshold has a jump immediately after a spike. Since the boundary depends on the previous firing epochs, the ISIs are not independent anymore. However, the distribution between two consecutive spikes, conditioned on the initial position of the threshold, is the same of that studied here. Hence, our results may represent a first step towards an understanding of the more complicated adapting-threshold models.\n\n\\subsection{Mathematical background} \\label{Sec1a}\nFPTs of diffusion processes to constant or time-dependent thresholds have been extensively studied in the literature. Explicit expressions for constant thresholds are available for the Wiener process \\cite{InverseGaussianBook,coxMiller}, for a special case of the Ornstein Uhlenbeck (OU) process \\cite{Ricciardi}, for the Cox-Ingersoll-Ross process \\cite{CapRic}, and for those processes which can be obtained from the previous through suitable measure or space-time transformations, see e.g. \\cite{Alili,CapRic,RicciardiW}. For most of the processes arising from applications and for time-varying thresholds, analytical expressions are not available.\nNumerical algorithms based on solving integral equations have been proposed in \\cite{BCCP,BNR,RicciardiNip,STZ3,Taillefumier,Telve},\nwhile approximations based on Monte-Carlo path-simulation methods in \\cite{GS, GSZ,Metzler}.\n\nA different approach to tackle the FPT problem consists in focusing directly on the two-sided boundary crossing probability (BCP), i.e. the probability that a process is constrained to be between two boundaries. If one of the boundary is set to $-\\infty$, the resulting one-sided BCP equals the survival probability of the FPT to the other boundary \\cite{Wang1997}. Explicit formulas for the BCP of a standard Wiener process for continuous and piecewise-linear thresholds are known (see \\cite{BorovkovNovikov,Novikovetal,Wang2001,Wang1997,Wang2007}). In general, the BCP of a diffusion process through an exponential decaying threshold is available only for those processes which can be expressed as a piecewise monotone functional of a standard Brownian motion. Examples are the OU process or the geometric Brownian motion with time dependent drift for specific parameter values \\cite{Wang2007}.\nThe simple but powerful idea is to approximate both one and two-sided curvilinear BCPs by similar probabilities for close boundaries of simpler form, namely $n$ piecewise-linear thresholds, whose computation of the BCP for Wiener is feasible. Under some mild assumptions, the approximated two-sided BCP converges to the original one when $n\\to\\infty$ \\cite{Wang2007}, with rate of convergence given in \\cite{BorovkovNovikov,Wang1997}.\n\nFor the exponential decaying threshold considered in this paper,\nthe convergence can be obtained by choosing piecewise linear thresholds approximating the curved boundary from above and below, with approximation accuracy given by their distance \\cite{Wang2007}. However, all the available formulas for the BCPs require either Monte-Carlo simulation methods or heavy numerical approximations.\n\nHere we consider a two-piecewise linear threshold as an approximation of the curvilinear boundary. Since $n=2$, the asymptotic convergence of the BCPs does not hold. However, we can derive analytical expression for the FPT density to the two-piecewise linear boundary, and use it as an approximation of the unknown FPT density. Four possible piecewise thresholds are proposed and optimized to minimize the distance to the original threshold.\n\n\\section{Model}\\label{Sec2}\nWe describe the membrane potential evolution of a single neuron by a Wiener process $X(t)$, starting at some initial value $x_0$. We assume $X(t)$ given as the solution to a stochastic differential equation\n\\begin{equation}\\label{model}\n\\left\\{\n\\begin{array}{l}\ndX(t)=\\mu dt + \\sigma dW(t),\\\\\nX(t_0)=x_0, \\qquad t>t_0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $W(t)$ is a standard (driftless) Wiener process. The drift $\\mu>0$ and the diffusion coefficient $\\sigma>0$ represent input and noise intensities, respectively. A spike occurs when the membrane potential $X(t)$ exceeds the exponentially decaying threshold\n\\begin{equation}\\label{b}\nb^*(t)=b_0+\\epsilon \\exp\\left[-\\lambda (t-\\delta_k)\\right]\n\\end{equation}\nfor the first time. Here, $\\delta_k$ denotes the time of the $k$th spike for $k>0$, and can be interpreted as a relative refractory period. We set $\\delta_0$ to be the starting time of the process, i.e. $\\delta_0=t_0$. The term $\\lambda$ represents the decay rate of the threshold, while $\\epsilon$ is interpreted as the amplitude of the time-dependent change in the boundary. After a spike, the membrane potential is reset to its resting position $x_0t_0: X(t)\\geq b(t)\\}.\n\\end{equation*}\nQuantities of interest are the probability density function (pdf) and the cumulative distribution function (cdf) of $T_b$, denoted by $f_{T_b}$ and $F_{T_b}$, respectively. Another relevant quantity is the two-sided BCP given by\n\\begin{equation*\n\\mathbb{P}_X(a,c,\\tau)=\\mathbb{P}\\left(a(t)t_0$ is fixed, boundaries $a(t)$ and $c(t)$ are real functions satisfying $a(t)\\tau)=1-F_{T_c}(\\tau),\n\\]\nwhich corresponds to the survival probability of $T_c$. For a standard Wiener process $W$, Wang and P\\\"{o}tzelberger \\cite{Wang2007} showed that, if the sequences of piecewise linear functions $a_n$ and $c_n$ converge uniformly to $a(t)$ and $c(t)$ on $[t_0,\\tau]$ respectively, then, for the continuity property of probability measure, it holds\n\\begin{equation*\n\\lim_{n\\to \\infty} P_W(a_n,c_n,\\tau)=P_W(a,c,\\tau).\n\\end{equation*}\nWhen $a(t)=-\\infty$ and $c(t)=b(t)$, the convergence of $\\mathbb{P}(T_{b_n}>\\tau)$ to $\\mathbb{P}(T_b>\\tau)$ can be obtained by choosing piecewise linear thresholds approximating $b(t)$ from above, denoted by $b_n^+(t)$, or from below, $b_n^-(t)$. That is, $b_n^+(t)\\geq b(t)$ and $b_n^-(t)\\leq b(t), \\ \\forall t\\in[t_0,\\tau]$, respectively. Since the considered curved boundary is convex, we have \\cite{Wang2001}\n\\begin{equation}\\label{updown}\n\\mathbb{P}_X(-\\infty,b_n^-,\\tau)\\leq \\mathbb{P}_X(-\\infty,b,\\tau)\\leq \\mathbb{P}(-\\infty,b_n^+,\\tau),\n\\end{equation}\ni.e.\n\\begin{equation*\n\\mathbb{P}(T_{b^+_n}\\leq \\tau)\\leq \\mathbb{P}(T_b\\leq \\tau)\\leq \\mathbb{P}(T_{b^-_n}\\leq \\tau).\n\\end{equation*}\nThe approximation accuracy is given by $\\mathbb{P}_X(-\\infty,b_n^+,\\tau)-\\mathbb{P}_X(-\\infty,b_n^-,\\tau)=F_{T_{b^-_n}}(\\tau)-F_{T_{b^+_n}}(\\tau)$, with bounds given in \\cite{BorovkovNovikov}. Obviously, the accuracy in the BCP increases when the distance between the two thresholds decreases.\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{FPTIF}\n\\caption{Schematic illustration of the single trial of a Wiener process in presence of exponentially decaying threshold $b(t)=b_0+\\epsilon\\exp(-\\lambda (t-\\delta_k))$, where $\\delta_k$ denotes the $k$th spike. The membrane potential starts in $x_0=0$ at time $\\delta_0:=t_0=0$, and it evolves until it hits the boundary for the first time. Then, a spike is generated, the voltage $X(t)$ is reset to its resting potential $x_0$, the threshold is reset to $b_0+\\epsilon$ and the evolution restarts. For the considered spiking generation rule, all ISIs are independent and identically distributed. Here the parameters are $\\mu=1, \\sigma^2=1, b_0=1, \\lambda=1$ and $\\epsilon=5$.}\n\\label{FigFPT}\n\\end{figure}\n\n\\section{FPT to continuous piecewise linear threshold}\\label{Method}\nThe transition density function of a standard Brownian motion in $x_1, x_2,\\ldots , x_n$ at time $t_1, t_2, \\ldots, t_n$, constrained to be below the absorbing threshold $c(t)$ defined by $n$ piecewise-linear threshold over $[t_0,t_n]$, is given in \\cite{Wang1997}. Extending that result to the case of a Brownian motion with drift $\\mu$ and diffusion coefficient $\\sigma$, starting in $x_0t_n)=\\int_{C_1}\\cdots\\int_{C_n}p_{c}(x_1,t_1;\\ldots;x_n,t_n|x_0,t_0)dx_1\\cdots dx_n,\n\\end{align}}}\nfor any Borel set $C_i\\subseteq (-\\infty,c_i), 1\\leq i \\leq n$. If $C_i=(-\\infty,c_i)$, then \\eqref{eq2.10} is equal to $\\mathbb{P}(T_c>t_n)$, and it holds\n\\begin{equation}\\label{fpt}\nf_{T_c}(t)=-\\frac{\\partial }{\\partial t_n} \\int_{-\\infty}^{c_1}\\cdots\\int_{-\\infty}^{c_n} p_{c}(x_1,t_1;\\cdots,x_n,t_n|x_0,t_0) dx_1\\cdots dx_n.\n\\end{equation}\n\nWhen $n=1$, the pdf $f_{T_c}$ is known. Since $X$ is a Wiener process with positive drift, the distribution of the FPT to $c(t)=\\alpha+\\beta (t-t_0)$ is inverse Gaussian,\n$T_{c}\\sim IG\\left[(\\alpha-x_0)\/(\\mu-\\beta), (\\alpha-x_0)^2\/\\sigma^2\\right]$, with pdf\n\\begin{equation}\\label{fpt3}\nf_{T_c}(t)=\\frac{\\alpha-x_0}{\\sqrt{2\\pi\\sigma^2(t-t_0)^3}}\\exp\\left(-\\frac{\\left[\\alpha-x_0-(\\mu-\\beta)(t-t_0)\\right]^2}{2\\sigma^2(t-t_0)}\\right),\n\\end{equation}\nmean $\\mathbb{E}[T_{c}]=(\\alpha-x_0)\/(\\mu-\\beta)$ and variance $\\textrm{Var}(T_{c})=(\\alpha-x_0)\\sigma^2\/(\\mu-\\beta)^3$ \\cite{InverseGaussianBook,coxMiller}. Note that the distribution of $T_c$ is the same of that of the FPT of a Wiener process with positive drift $\\mu-\\beta$ to a constant threshold $c(t)=\\alpha$. In general, the approximation of $F_{T_b}$ by $F_{T_c}$ when $n=1$ is too rough. However, when $\\lambda$ is very small, $\\exp(-\\lambda t)\\approx 1-\\lambda t$, yielding $b(t)\\approx b_0+\\epsilon-\\lambda \\epsilon t$. Hence, $T_b$ can be approximated by $T_c$ with $\\alpha=b_0+\\epsilon$ and $\\beta=-\\lambda \\epsilon$.\n\nSince we approximate $b(t)$ by means of a continuous two-piecewise linear threshold, we denote by $\\tilde b$ the linear threshold $c(t)$ when $n=2$. We have\n\\begin{equation}\\label{St}\n\\tilde{b}(t)=\\tilde b_1(t)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\\tilde b_2(t) \\mathbbm{1}_{\\{t>t_1\\}}=\\left\\{\n\\begin{array}{ll}\n\\alpha_1+\\beta_1 (t-t_0) & \\textrm{ if } t_0\\leq t\\leq t_1\\\\\n\\alpha_2+\\beta_2(t-t_1) & \\textrm{ if } t> t_1\n\\end{array}\n\\right. ,\n\\end{equation}\nwhere $\\mathbbm{1}_A$ denotes the indicator function of the set $A$ and $\\alpha_1,\\alpha_2,\\beta_1,\\beta_2\\in \\mathbb{R}$. Throughout the paper, we set $\\alpha_2=\\alpha_1+\\beta_1 (t_1-t_0)$ to guarantee the continuity of $\\tilde b(t)$. This allows to provide analytical expressions of \\eqref{eq2.9} and \\eqref{fpt}, which we use as an approximation of $f_{T_b}$. In particular, we have\n\\begin{align}\\label{fpt2}\n\\nonumber\\mathbb{P}(T_{\\tilde{b}}t_1)\\\\\n\\nonumber=&\\ \\mathbb{P}(T_{\\tilde{b}_1} <\\min(t_1,t)) + \\int_{-\\infty}^{\\tilde{b}(t_1)} \\mathbb{P}(T_{\\tilde{b}_2} t_1\\}}\n\\frac{1}{\\sqrt{2\\pi\\sigma^2(t-t_0)^3}}\\exp\\left(-\\frac{(\\alpha_2-x_0-(\\mu-\\beta_2)(t-t_1)-\\mu(t_1-t_0))^2}{2\\sigma^2(t-t_0)}\\right)\\\\\n\\nonumber&\\times \\left\\{[\\alpha_2-x_0-\\beta_2(t_1-t_0)]\\Phi\\left(\\frac{(\\alpha_2-x_0-\\beta_2(t_1-t_0))\\sqrt{(t-t_1)}}{\\sqrt{\\sigma^2(t_1-t_0)(t-t_0)}}\\right)\\right.\\\\\\nonumber& \\left.-(\\alpha_2+x_0-\\beta_2(t_1-t_0)-2\\alpha_1\n\\exp\\left(-\\frac{2(t-t_1)(\\alpha_1-x_0)(\\alpha_2-\\alpha_1-\\beta_2(t_1-t_0))}{\\sigma^2(t_1-t_0)(t-t_0)}\\right)\\right.\\\\& \\times\\left.\\Phi\\left(\\frac{(\\alpha_2+x_0-\\beta_2(t_1-t_0)-2\\alpha_1)\\sqrt{(t-t_1)}}{\\sqrt{\\sigma^2(t_1-t_0)(t-t_0)}}\\right)\\right\\}\n\\end{align}}This result extends that for a driftless Brownian motion, see e.g. \\cite{Abundo,Scheike}.\nAs expected, setting $\\alpha_1=\\alpha_2=\\alpha$ and $\\beta_1=\\beta_2=\\beta$ yields the pdf of the FPT of a Wiener process to a linear threshold $c(t)=\\alpha+\\beta(t-t_0)$. By definition, the first two moments and variance of $T_{\\tilde b}$ are given by\n\\begin{equation}\\label{EVT}\n\\mathbb{E}[T_{\\tilde b}]=\\int_0^\\infty tf_{T_{\\tilde b}}(t)dt, \\qquad\n\\mathbb{E}[T^2_{\\tilde b}]=\\int_0^\\infty t^2 f_{T_{\\tilde b}}(t)dt, \\qquad \\textrm{Var}[T_{\\tilde b}]=\\mathbb{E}[T_{\\tilde b}^2]-\\mathbb{E}[T_{\\tilde b}]^2,\n\\end{equation}\nand can be numerically computed.\n\n\\section{Parameter estimation}\\label{Sec4}\n\\subsection{Parameter estimation of the piecewise-linear threshold}\\label{Sec4a}\nThe primary aim of this paper is the approximation of the FPT distribution (and relevant statistics) for a curved boundary $b(t)$, by means of the FPT distribution for a continuous two-piecewise linear threshold $\\tilde b(t)$. As discussed in Section \\ref{Sec2}, the quality of the approximation improves when the distance between $\\tilde b$ and $b$ decreases.\n\nDenote by $\\theta=(\\alpha_1,\\beta_1,\\beta_2,t_1)$ the parameters of $\\tilde b$ in \\eqref{St}, with $\\alpha_2=\\alpha_1+\\beta_1(t_1-t_0)$. We are interested in determining the estimator $\\hat\\theta$ which minimizes $|\\tilde b(t)-b(t)|$ on $[\\tau_0,\\tau_*] $, with $t_0<\\tau_0t)=\\mathbb{P}(X(s)b_0$, it follows that\n\\[\n\\mathbb{P}(X(t)\\geq b(t)) \\leq \\mathbb{P}(T_b\\leq t)\\leq \\mathbb{P}(T_{b_0} \\leq t),\n\\]\nwith $T_{b_0}\\sim IG((b_0-x_0)\/\\mu, (b_0-x_0)^2\/\\sigma^2)$. Since $X$ is a Wiener process, $X(t)\\sim N(\\mu t, \\sigma^2 t)$. Then, we choose $\\tau_0$ and $\\tau_*$ such that\n\\[\n\\mathbb{P}(T_{b_0}\\leq\\tau_0)=0.005, \\qquad\n\\mathbb{P}(X(\\tau_*)\\geq b(\\tau_*))=0.995,\n\\]\nyielding the desired probability \\eqref{mah}.\n\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{threshold}\n\\caption{Curved threshold $b(t)$ (continuous line) and four proposed approximating piecewise-linear thresholds: $b_+(t)$ from above (dashed lines) ; $b_-(t)$ from below (dashed-dotted line); $b_\\textrm{betw}(t)$ which is equidistant from $b_+$ and $b_-$ (gray dashed line); $b_\\textrm{free}(t)$ with no restrictions (gray continuous line). For each type of linear threshold, the best approximation is given by the line minimizing a function of the distance from $b$ on $[t_0=0,\\tau]$ (left figure) and on $[\\tau_0,\\tau_*]\\subseteq [t_0=0,\\tau]$ (right figure). As discussed in Section \\ref{Sec4a}, a shorter time interval provides a better approximation of $b$.}\n\\label{thresh}\n\\end{figure}\nThroughout the paper, we consider four possible continuous two-piecewise linear boundaries on $[\\tau_0,\\tau_*]$, as illustrated in Fig. \\ref{thresh}:\n\\begin{enumerate}\n\\item Threshold $b_+$ approximating $b$ from above on $[\\tau_0,\\tau_*]$, passing through $(\\tau_0,\\break b(\\tau_0))$, $(t_1,b(t_1))$ and\n $(\\tau_*,b(\\tau_*))$,\n\\begin{equation*}\\label{th2}\nb_+(t)=b(\\tau_0)+\\frac{b(t_1)-b(\\tau_0)}{t_1-\\tau_0}(t-\\tau_0)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\n\\frac{b(\\tau_*)-b(t_1)}{\\tau_*-t_1}(t-t_1)\\mathbbm{1}_{\\{t>t_1\\}},\n \\end{equation*}\n i.e.\n \\[\n \\alpha_1=b_+(t_0), \\quad \\beta_1= \\frac{b(t_1)-b(\\tau_0)}{t_1-\\tau_0}, \\quad \\beta_2=\\frac{b(\\tau_*)-b(t_1)}{\\tau_*-t_1}(t-t_1).\n \\]\nDue to the assumptions, for given $\\tau_0$ and $\\tau_*$, $t_1$ is the only unknown quantity.\n\\item Threshold $b_-$ approximating $b$ from below on $[\\tau_0,\\tau_*]$. We assume that $b_-$ is tangent to $b(t)$ in both $\\tilde t_1$ and $\\tilde t_2>\\tilde t_1$, with $t_1$ intersection time point of the two tangent lines\n\\[\ny_i(t)=b(\\tilde t_i)-\\lambda \\epsilon \\exp(-\\lambda \\tilde t_i)(t-\\tilde t_i),\n\\]\nfor $i=1,2$. Setting $y_1(t_1)=y_2(t_1)$, we get\n\\[\nt_1=\\frac{\\exp(-\\lambda \\tilde t_1)[1+\\lambda \\tilde t_1]-\\exp(-\\lambda \\tilde t_2)[1+\\lambda \\tilde t_2]}{\\lambda[\\exp(-\\lambda \\tilde t_1)-\\exp(-\\lambda \\tilde t_2)]}.\n\\]\nThen, the desired threshold $b_-(t)$ is\n\\[\nb_-(t)=y_1(\\tilde t_1)+\\frac{y_1(t_1)-y_1(\\tilde t_1)}{t_1-\\tilde t_1}(t-\\tilde t_1)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\\frac{y_2(\\tilde t_2)-y_2(t_1)}{\\tilde t_2-t_1}(t- t_1)\\mathbbm{1}_{\\{t>t_1\\}},\n\\]\nwith\n\\[\n\\alpha_1=b_-(t_0),\\quad \\beta_1=\\frac{y_1(t_1)-y_1(\\tilde t_1)}{t_1-\\tilde t_1}, \\quad\n\\beta_2=\\frac{y_2(\\tilde t_2)-y_2(t_1)}{\\tilde t_2-t_1}.\n\\]\nFor fixed $\\tau_0$ and $\\tau_*$, the unknown parameters are $\\tilde t_1$ and $\\tilde t_2$.\n\\item Threshold $b_\\textrm{betw}(t)$ constrained to be between $b_+(t)$ and $b_-(t)$ on $[\\tau_0,\\tau_*]$, i.e.\\break $b_-(t)\\leq b_\\textrm{betw}(t)\\leq b_+(t)$.\n\\item Threshold with no constraints, denoted by $b_\\textrm{free}(t)$.\n\\end{enumerate}\nDenote by $\\hat\\theta_+, \\hat\\theta_-,\\hat\\theta_\\textrm{betw}$ and $\\hat\\theta_\\textrm{free}$ the estimators of $\\theta$ from the boundaries $b_+, b_-,b_\\textrm{betw}$ and $b_\\textrm{free}$, respectively.\nFrom \\eqref{updown}, it follows that the best approximation of $\\mathbb{P}_X(-\\infty,b,\\break\\tau)$ is obtained when the distance between $b_+$ and $b_-$ is minimized. For this reason, we define $\\hat\\theta_+$ and $\\hat\\theta_-$ as the estimators minimizing the area of the squared distance between the two boundaries, i.e.\n\\begin{equation*\n(\\hat\\theta_+,\\hat\\theta_-)=\\arg\\min_{(\\theta_+,\\theta_-)}\\left[ \\int_{\\tau_0}^{\\tau{^*}} |b_+(t)-b_-(t)|^2dt\\right],\n\\end{equation*}\nwith $\\hat\\theta_+$ and $\\hat\\theta_-$ satisfying the conditions $b_+(t)>b(t)$ and $b_-(t)x_0$, we have\n\\begin{eqnarray}\n\\label{ET} \\widehat{\\mathbb{E}[T_b]}&=& \\frac{b_0}{\\mu}+\\frac{\\epsilon}{\\mu}\\exp\\left(\\frac{b_0\\left(\\mu-\\sqrt{\\mu^2+2\\lambda\\sigma^2}\\right)}{\\sigma^2}\\right),\\\\\n\\nonumber \\widehat{\\textrm{Var}(T_b)}&=&\\frac{b_0\\sigma^2}{\\mu^3}+\\frac{\\sigma^2\\epsilon}{\\mu^3}(b_0-1)\\\\\n&+&\\frac{2\\epsilon}{\\mu^2}\\left(\\frac{\\mu b_0}{\\sqrt{\\mu^2+2\\lambda\\sigma^2}}+\\frac{\\sigma^2}{2\\mu}-\\theta_0\\right)\\exp\\left(\\frac{b_0\\left(\\mu-\\sqrt{\\mu^2+2\\lambda\\sigma^2}\\right)}{\\sigma^2}\\right).\\quad\n\\label{VT}\n\\end{eqnarray}\nWe denote by $\\hat\\phi_{\\textrm{ME}}^\\epsilon$ the moment estimator of $\\phi$ obtained from \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small.\n\n\\section{Simulation study}\n\n\\subsection{Monte Carlo simulations} We simulate FPTs of the Wiener process $X$ to $b(t)$ as described in \\cite{LindnerLongtin,ReviewSac}. Applying the Euler-Maruyama scheme to the stochastic differential equation \\eqref{model}, we generate realizations of $X$, denoted by $x_i:=X(s_i)$, at discrete times $s_i=i \\Delta s, i\\geq1$. We set $X_0=x_0=0$ and $\\Delta s=0.001$ as time step. To avoid the risk of not detecting a crossing of the boundary due to the discretization of the sample path, at each iteration step we compute the probability that the bridge process $X^{[s_i,s_{i+1}]}=\\left\\{X_s^{[s_i,s_{i+1}]},s\\in [s_i,s_{i+1}]\\right\\}$, originated in $x_iu_i$. In this case, the mid-point $(s_i+s_{i+1})\/2$ is chosen as simulated FPT. Samples of size $100$ are simulated for different values of $\\sigma^2, \\lambda$ and $\\epsilon$ when $b_0=1$ and $\\mu=1$. In particular, we consider $\\sigma^2=0.2, 0.4, 1$; $\\epsilon=0.05, 0.1, 0.2, 1, 5, 10$ and $\\lambda=0.02, 0.04, 0.08, 0.15, 0.30, 0.60, 1.00, 3.00, 5.00, 10.00$. These parameter values are chosen to cover and extend the cases of small values of $\\epsilon$ considered in \\cite{LindnerLongtin,Urdapilleta}. Finally, for each value of $\\sigma^2,\\epsilon$ and $\\lambda$, we repeat simulation of data set 1000 times, obtaining 1000 statistically indistinguishable and independent trials.\n\n\\subsection{Set up} In the simulations we are mainly concerned to illustrate the performance of our method under the assumption that the threshold $b(t)$ is known, i.e. $b_0$, the rate $\\lambda$ and the amplitude $\\epsilon$ are given. Two scenarios are considered: both $\\mu$ and $\\sigma^2$ are known; no information about the parameter of the Wiener process is given.\nIn the first case it is of interest to evaluate the error in the estimation of mean, variance, CV and cdf of $T_b$ by comparing theoretical \\eqref{EVT} and empirical firing statistics. When $\\epsilon$ is small, a further comparison with \\eqref{ET} and \\eqref{VT} is carried out. To measure the error in the estimation of $F_{T_b}$, we consider the relative integrate absolute error ($R_\\textrm{IAE}$), defined as\n\\begin{equation}\\label{riae}\nR_{\\textrm{IAE}}(\\hat F_{T_b})=\\frac{\\int_0^{\\infty} | \\hat F_{T_b}(t)-F_{T_b}(t)| dt}{\\mathbb{E}[T_b]}.\n\\end{equation}\nWe replace the unknown quantities $F_{T_b}$ and $\\mathbb{E}[T_b]$ by their empirical estimators, defined by $F_n(t)=\\frac{1}{n}\\sum_{i=1}^n \\mathbbm{1}_{\\{T_{b_i}\\leq t\\}}$ and $\\bar t=\\sum_{i=1}^n T_{b_i}\/n$, respectively. Both empirical quantities are based on $n=1000 000$ simulated FPTs, ensuring the closeness to the theoretical counterparts by the law of large numbers. This first scenario is meant to understand the goodness of our approximation through simulations.\n\nAnother relevant question is the performance of the MLEs and MEs of $\\mu$ and $\\sigma^2$, as described in Section \\ref{Sec4b}. To compare different estimators, we use the relative mean error $R_\\textrm{ME}$ to evaluate the bias and the relative mean square error $R_{\\textrm{MSE}}$, which incorporates both the variance and the bias. They are defined as the average over the $1000$ repetitions of the quantities\n\\[\nE_\\textrm{rel}(\\hat\\mu)=\\frac{\\hat\\mu-\\mu}{\\mu}, \\qquad E_\\textrm{rel sq}(\\hat\\mu)=\\frac{(\\hat\\mu-\\mu)^2}{\\mu^2},\n\\]\nand likewise for $\\sigma^2$.\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{allb}\\\\%\\includegraphics[width=1.3\\textwidth]{all}\n\\includegraphics[width=1.0\\textwidth]{all3b}\\\\\n\\includegraphics[width=1.0\\textwidth]{all5}\n\\caption{Mean (left panels), variance (central panels) and CV (right panels) of the FPT $T_b$\n as a function of the decay rate $\\lambda$ of the threshold for small values of the amplitude $\\epsilon$ when $\\mu=1$. Top panels: $\\sigma^2=0.2$. Central panels: $\\sigma^2=0.4$. Bottom panels: $\\sigma^2=1$. Empirical quantities from simulations (symbols), theoretical quantities given by \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$ (solid lines), and theoretical quantities \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small (solid gray lines). Also shown are the firing statistics of $T_b$ when $\\epsilon=0$ (horizontal dashed lines).}\n\\label{Figall}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{all2}\n\\caption{Mean (left panel), variance (central panel) and CV (right panel) of the FPT $T_b$\n as a function of the decay rate $\\lambda$ of the threshold for large values of the amplitude $\\epsilon$ when $\\mu=1$ and $\\sigma^2=0.2$. Empirical quantities from simulations (symbols), theoretical quantities given by \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$ (solid lines), and theoretical quantities \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small (solid gray lines). Also shown are the firing statistics of $T_b$ when $\\epsilon=0$ (horizontal dashed lines).}\n\\label{Figall2}\n\\end{figure}\n\n\n\\subsection{Theoretical results for cdf and firing statistics of $T_b$} In Fig. \\ref{Figall} are reported theoretical and empirical means, variances and CVs of $T_b$ as a function of the rate $\\lambda$, for small values of the amplitude $\\epsilon$ and for $\\sigma^2=0.2, 0.4$ and $1$. The given theoretical quantities are obtained from \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$. Note how the mean of $T_b$ does not depend on $\\sigma^2$, as it also happens for a linear threshold, while both variance and CV increase with growing $\\sigma^2$. We refer to \\cite{LindnerLongtin} for a detailed discussion on other qualitative features of the firing statistics, e.g. monotonic decrease on the mean with growing $\\lambda$, existence of a minimum value for the variance, etc. What is relevant to emphasize is the excellent fit of the firing statistics provided by our method for any $\\lambda$, and for both small (cf. Fig. \\ref{Figall}) and large (cf. Fig. \\ref{Figall2}) values of $\\epsilon$. When $\\epsilon$ is small, our theoretical firing statistics are at least as good as those in \\cite{LindnerLongtin, Urdapilleta}, outperforming them when $\\epsilon$ grows. The firing statistics of $T_{b_\\textrm{betw}}$ are almost identical to those of $T_{b_\\textrm{free}}$, while those of $T_{b_+}$ and $T_{b_-}$ are slightly different for increasing $\\epsilon$. This can be seen in Fig. \\ref{Figriae}, left panel, where the percentages of the $R_\\textrm{IAE}(\\hat F_T)$ for the four proposed estimators are given. As expected, the best approximation of $F_{T_b}$ is provided by $F_{T_{b_\\textrm{free}}}$, since $b_\\textrm{free}$ is the only threshold whose parameters are obtained from a non-constrained optimization problem. The performance of the estimators gets worse for large $\\sigma^2$ and $\\epsilon$. The highest error is observed for the value of $\\lambda$ that minimizes the variance of $T_b$. However, all errors are smaller than $2\\%$, confirming the good performance of the proposed estimators.\n\n\\subsection{Parameter estimation of $(\\mu,\\sigma^2)$}\nWe have seen that $T_{b_\\textrm{free}}$ yields the best approximation of $T_b$ in terms of both cdf and firing statistics. For this reason, we limit our study to the estimators $\\hat\\phi$ based on $b_\\textrm{free}$. In Fig. \\ref{Figphi1} the $R_\\textrm{ME}$ and the $R_\\textrm{MSE}$ of $\\hat\\mu$ and $\\hat\\sigma^2$ are reported. As expected, the MLE provides the best estimate of $\\phi$, while both MEs are acceptable only for small values of $\\epsilon$. The performance of $\\hat\\mu$ is highly satisfactory, with $R_\\textrm{ME}(\\hat\\mu)$ smaller than $0.5\\%$, and $R_\\textrm{MSE}(\\hat\\mu)<0.2\\%$. Larger but still good $R_\\textrm{ME}$ and $R_\\textrm{MSE}$ are observed for $\\hat\\sigma^2$. The performance of $\\hat\\phi_\\textrm{MLE}$ gets worse for growing $\\sigma^2$, as shown in Fig. \\ref{Figphi2}. However, except the $R_\\textrm{ME}(\\hat\\sigma^2)$ for large values of $\\epsilon$, all errors are between $0$ and $2-3\\%$. Two last remarks should be done: first, the $R_\\textrm{MSE}$ of $\\hat\\mu$ for small values of $\\epsilon$ approaches the corresponding values of $\\sigma^2$.\nSecond, $R_\\textrm{MSE}(\\hat\\sigma^2)$ seems not to depend on $\\lambda, \\epsilon$ and $\\sigma^2$, but to be equal to $2\\%$. This error decreases when increasing the sample size. For example, the $R_\\textrm{MSE}(\\hat\\sigma^2)\\approx 1\\%$ when $n=200$ (results not shown).\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{riaeb}\n\\caption{$R_{\\textrm{IAE}}(\\hat F_{T_b})$ (in percentage) given by \\eqref{riae} for different values of $\\lambda$ and $\\epsilon$ when $\\mu=1$. Left panel: $R_{\\textrm{IAE}}(\\hat F_{T_b})$ from threshold $b_\\textrm{free}$ (circles), $b_-$ (triangles), $b_+$ (rhombuses) and $b_\\textrm{betw}$ (gray circles) when $\\epsilon=1$ and $\\sigma^2=0.2$. Right panel: $R_\\textrm{IAE}(\\hat F_{T_{b_\\textrm{free}}})$ for $\\sigma^2=0.2$ (circles), $\\sigma^2=0.4$ (triangles) and $\\sigma^2=1$ (gray circles). The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.}\n\\label{Figriae}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{est_phi}\n\\caption{Dependence of $R_\\textrm{ME}(\\hat\\mu), R_\\textrm{MSE}(\\hat\\mu), R_\\textrm{ME}(\\hat\\sigma^2)$ and $R_\\textrm{MSE}(\\hat\\sigma^2)$ (average over $1000$ simulations) on $\\lambda$ and $\\epsilon$ when $X$ is a Wiener process with $\\mu=1$ and $\\sigma^2=0.2$. Different estimators of $\\phi=(\\mu,\\sigma^2)$ are considered: maximum likelihood estimator $\\hat\\phi_\\textrm{MLE}$ (solid lines with triangles), moment estimator $\\hat\\phi_\\textrm{ME}$ (dashed lines with circles) and moment estimator from \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small, $\\hat\\phi_{ME}^\\epsilon$ (gray solid lines with gray circles). The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.\n}\n\\label{Figphi1}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{est_phi2}\n\\caption{Dependence of $R_\\textrm{ME}(\\hat\\mu), R_\\textrm{MSE}(\\hat\\mu), R_\\textrm{ME}(\\hat\\sigma^2)$ and $R_\\textrm{MSE}(\\hat\\sigma^2)$ (average over $1000$ simulations) on $\\lambda, \\epsilon$ and $\\sigma^2$ when $X$ is a Wiener process with $\\mu=1$ and $\\sigma^2$ equal to $0.2$ (solid lines with circles), $0.4$ (dashed lines with triangles) and $1$ (gray solid lines with gray circles). Here only the maximum likelihood estimator $\\hat\\phi_\\textrm{MLE}$ of $\\phi=(\\mu,\\sigma^2)$ is considered. The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.}\n\\label{Figphi2}\n\\end{figure}\n\n\\section{Discussion} As a consequence of the recent increasing interest towards adapting-threshold models for the description of the neuronal spiking activity, a need of suitable mathematical tools to deal with hitting times of diffusion processes to time-varying thresholds arises. The mathematical literature on the FPT problem is rich and extensive. Unfortunately, analytical solutions are not available even for a problem as simple (compared to others) as the one considered here, i.e. Wiener process to an exponentially decaying threshold. The closest result in this direction is represented by the work of Wang and P\\\"{o}tzelberger, who provide an explicit expression which should then be evaluated through Monte-Carlo simulations. The idea behind the works of Lindner and Longtin and of Urdapilleta, was to simplify some mathematical difficult equations arising from the study of the FPT by linearizing them in $\\epsilon$, the amplitude of the decaying threshold. As a consequence, the quality of the approximation rapidly decreases when $\\epsilon$ increases.\n\nThe method proposed here has no restriction on the parameter of the thresholds and it is based on the simple idea of replacing the boundary by a continuous two-piecewise linear threshold. This allows us to derive the analytical expression of the FPT density to the two-piecewise threshold, and to use it to approximate the desired distribution. To some extent, the presence of two linear thresholds can be considered as a second order approximation of the problem.\n\nNumerical simulations show a good performance of the proposed method both when computing the main firing statistics, such as means, variances and CVs, and when calculating the FPT distribution. Different approximating thresholds have been proposed. We suggest choosing the one minimizing the distance with the curvilinear threshold and to restrict the interval where to perform the optimization as described in the paper. Among the estimators of the drift and diffusion coefficients of the Wiener process, we suggest applying MLE which always estimates the parameters reasonably well.\n\nThe method proposed here may yield several interesting developments. First of all, it can be used to characterize the firing statistics of the Wiener process to the exponential decaying threshold, extending the previous considerations obtained for small values of $\\epsilon$. Then, it may be extended to the case of a Wiener process with an adapting decaying threshold, as suggested in the introduction. Finally, our results may also be applied to all those processes which can be expressed as a piecewise monotone functional of a standard Brownian motion \\cite{Wang2007}, as well as to Wiener processes with time-varying drift \\cite{LindnerLongtin,Molini2011}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{Intro}\n\nIn their pioneering papers \\cite{wick,cutk}, Wick and Cutkosky (W-C) have\nfound the solutions of the Bethe-Salpeter (BS) equation \\cite{bs} for\ntwo scalar particles interacting by the exchange of a massless scalar\nparticle. In addition to the states which, in the non-relativistic\nlimit, reproduce the spectrum of the Schr\\\"odinger equation with the\nCoulomb potential, there was found another set of solutions which do\nnot have any non-relativistic counterparts. These solutions were\ncalled ``abnormal''.\nTheir discovery triggered the discussion as to whether they do\nindicate a mathematical inconsistency of the W-C model or\nof the BS equation, or whether they represent new physical systems,\nwhose existence does not contradict any physical principles, \nalthough they are not covered by the Schr\\\"odinger equation.\nIn the latter case, they might provide examples of relativistic\nsystems which could exist in nature, but which would not be\ndescribed by continuous extensions of non-relati\\-vis\\-tic quantum\nmechanics. A thorough discussion of this issue can be found in Ref.\n\\cite{nak69} (sections 6 and 8).\n\nOne would hope that a complementary lighting to the above questioning\nmight come from experimental data. Unfortunately the conditions of the\nemergence of abnormal states are not easy to realize.\nConsidering the W-C model as a simplified model of QED,\nabnormal states would appear as highly excited states for values of\nthe fine\nstructure constant $\\alpha$ above $0.5\\div 1$. Such values might be\nreached with the\naid of heavy ions and dedicated electron-ion scattering experiments\nmight be envisaged. However, to have a clear experimental distinction\nof abnormal bound states from the ionization threshold, one actually\nwould need to increase the values of $\\alpha$ up to $4\\div 5$, which\nthen further reduce the probability of an experimental success.\nAnother possibility is an analogy of the model with hadron dynamics,\nwhere hadrons mutually interact by means of the exchange of light\nparticles, like the pions, and where the coupling constants might\nlie in the range of values needed for the existence of abnormal\nstates. However, here, the exchanged particles being massive,\ndrastic changes occur with respect to the massless case: the\ninteraction forces become of short-range and one realizes that\nabnormal states are produced only with very small mass values of\nthe exchanged particle, much smaller than the pion mass. \nThe framework of Quantum Chromodynamics, where quarks\nmutually interact by means of exchange of massless gauge particles,\nthe gluons, with sufficiently strong forces, might provide another\ndomain to search for possible evidences of abnormal solutions.\n\n\nIn the absence of any direct experimental indication about the\nexistence or nonexistence of abnormal states, one is entitled to\nexplore all possible theoretical paths that might provide\ncomplementary information about their properties. From this point\nof view, we have found that an analysis of the Fock-space content\nof the abnormal, as well as normal, states would be of great help.\nThe BS amplitude allows one to extract the wave function related\nto the two-body sector of the Fock space \\cite{cdkm}. \nIts norm, which is positive and bounded by 1, is then\ninterpreted as the weight of that sector in the whole Fock space.\n\nComplementary information to the above analysis comes from the\nknowledge of the electromagnetic form factors, assuming that one\nof the massive constituents of the bound state is charged. Their\nasymptotic behavior qualitatively probes the compositeness of the\nstates: a rapid decrease would be the signature of a many-body\nstructure \\cite{matvmurtavk,brodsfarr,radyush}. \n\nOur calculations, as well as the results of ref. \\protect{\\cite{dshvk}},\nshow that, in the window of allowed coupling constants, the normal\nsolutions are essentially dominated by the two-body sector of the Fock\nspace. \nWe will show in the present work that, on the contrary, \nthe abnormal solutions have a two-body contribution that vanishes in\nthe limit of zero binding energies and remains small (less than\n10\\%) in all its domain of existence.\nThey are therefore dominated by the many-body sectors, composed of the\ntwo massive constituents and of several massless exchange\nparticles. This feature explains why the abnormal\nsolutions disappear from the spectrum in the non-relativistic\nlimit, the latter being formulated in the two-body sector alone,\nwhile the other sectors, containing massless particles, are by\nessence relativistic.\n\nThe asymptotic behaviors of the form factors also corroborate the\nabove conclusions. The form factors of the abnormal solutions\nasymptotically decrease, for spacelike momenta, faster, by factors\nof the order of $10^{3}$, than those of the normal solutions.\nAlso, the transition form factors between normal and abnormal\nsolutions display global suppressions, by factors of $5\\div 10$, with\nrespect to the normal-normal or abnormal-abnormal transition form\nfactors, signalling a different nature of the normal and abnormal\nsolutions.\n\nThese results suggest that the abnormal solutions might correspond\nto states called ``hybrids'' in the literature. In the present model,\nthey are dominated\nby Fock space sectors containing two massive constituents and several\nor many massless constituents, corresponding to the exchanged-particle\nfields.\nThey could be considered as the Abelian scalar analogs of the QCD\nhybrids, which, in the mesonic sector, are dominated by their\ncoupling to the set of fields made of a quark, an antiquark and one or\nseveral gluon fields.\n\nFinally, the question of the validity of the ladder approximation\nof the model, because of the necessity of having large values of the\ncoupling constant to create abnormal states, still remains an open issue.\n\nThe plan of the paper is the following. Sec. \\ref{def} is devoted to an\nintroductory definition of the BS amplitude and of the Fock-space sectors. \nIn Sec. \\ref{WCsol}, the properties of the solutions of the\nW-C model are displayed and some solutions are found numerically. \nIn Sec. \\ref{FFs} the elastic and transition electromagnetic form factors\nare expressed through the BS amplitudes and are calculated numerically,\nwith special emphasis put on their asymptotic behavior. Concluding remarks\nfollow in Sec. \\ref{concl}. \nThree appendices give technical details\nabout some of the formulas used in the main text. Preliminary results\nof the present study were presented in Ref. \\cite{LC2019}.\n\n\\par\n\\section{Fock space sectors} \\label{def}\n\nThe BS amplitude, satisfying the BS equation, is defined as\n\\begin{equation}\\label{bs1}\n\\Phi(x_1,x_2;p)\\equiv \\langle 0 |T[\\phi_1(x_1)\\phi_2(x_2)]|p\\rangle,\n\\end{equation}\nwhere $\\phi_a(x)$ ($a=1,2)$ are Heisenberg field operators,\n$T$ means time ordering, $|p\\rangle$ is the state vector of the\nbound system and $\\langle 0 |$ is the vacuum state vector.\nSince the amplitude $\\Phi(x_1,x_2;p)$ depends on two 4D variables,\n$x_1$ and $x_2$, it is usually called ``two-body'' BS amplitude,\nthough this terminology, to some extent, is misleading.\nAsking the questions ``what is the content of a system?'' or ``is it\ntwo-body or many-body?'' requires that we analyze the state\nvector $|p\\rangle$ of this system, entering in the matrix element\n(\\ref{bs1}), by decomposing it onto the states $| n\\rangle$ with\ndefinite numbers $n$ of particles (the Fock sector decomposition),\nschematically:\n\\begin{equation}\\label{p}\n|p\\rangle=\\sum_{n=2}^{\\infty}\\psi_n^{}|n\\rangle,\n\\end{equation}\nand studying the contributions of the two-body component $\\psi_2$,\nthe three-body component $\\psi_3$, etc., in the full normalization\nintegral. The answer to the above questions depends on which component\n(or sum of components) is dominant. It should be mentioned that the state vector $|p\\rangle$\n is usually defined on a $t$-constant plane in the 4D space. There are however some advantadges to choose the so called\nlight-front plane \n$t+z=0$ (or light-front plane of general orientation, see \\cite{cdkm}). In this case, the corresponding Fock components \n$\\psi_n$ are called the light-front wave functions. The two-body light-front wave function $\\psi_2$ is related to the BS \namplitude (\\ref{bs1}) by eq. (\\ref{lfwf}) from \\ref{appN2}.\n\\par\n\nIn the W-C model, in the ladder approximation, the\nFock decomposition can contain two constituent (massive) particles\nand any number of exchange (massless) particles. The state\n$|2\\rangle$ (two-body sector) contains two constituents only, the\nstate $|3\\rangle$ (three-body sector) contains two constituents and\none exchange particle, the state $|n\\rangle$ ($n$-body sector)\ncontains two constituents and $(n-2)$ exchange particles, etc. \nAssuming that the state vectors $|n\\rangle$ ($n=2,3,\\ldots$) are\nnormalized to unity, the state vector $|p\\rangle$ is then normalized\nas\n\\begin{equation}\\label{eq2}\n\\langle p|p\\rangle=\\sum_{n=2}^{\\infty}N_n=1,\n\\end{equation}\nwhere, schematically, $N_n=\\int |\\psi_n|^2\\ldots$ \nis the contribution of the $n$-body Fock sector\n(see eq. (\\ref{N2}) for the exact definition of $N_2$).\nIn practice, knowing the BS amplitude $\\Phi(x_1,x_2;p)$ we are able to find $N_2$ only. \nThe calculation of $N_2$ from the BS amplitude is presented in\n\\ref{appN2}. If it is dominant, this would mean that the\ncontribution of the other sectors, containing exchange\nparticles, is small. The limiting case, when it is enough to keep the\ntwo-body state only (the case corresponding to $N_2=1$), whereas the\nstates containing exchange particles can all be omitted, is realized\nin non-relativistic systems. On the contrary, when the two-body\ncontribution $N_2$ is small, the system is dominated by \ntwo constituents with an indefinite number of exchange massless \nparticles, whose contribution $\\sum_{n=3}^{\\infty}N_n$ is close to 1.\n\\par\n\nFor the normal solutions of the W-C model (in the\nequal-mass case), the above analysis\nhas been made in Ref. \\cite{dshvk}. It was found that for small binding\nenergies the two-body (constituent) sector dominates, as expected. When\nthe binding energy increases (i.e., the total mass $M$ decreases), the\ntwo-body contribution $N_2$ decreases in parallel. However, it still\ndominates and as $M\\to 0$, it tends, in this model, to 64\\%.\nThat is, the sectors $|n\\rangle$ with $n\\geq 3$ contribute in total\nto 36\\% of the total normalization of the normal state vector.\nIn the present paper, we will carry out the same analysis for the\nabnormal states.\n\n\n\\section{Wick-Cutkosky solutions}\\label{WCsol}\n\nThe BS equation \\cite{bs} for the amplitude (\\protect{\\ref{bs1}})\ncontaining two spinless fields, \nrestricted to the equal-mass case $m_1=m_2=m$, reads, in momentum space,\n\\begin{eqnarray}\\label{bs}\n\\Phi(k,p)&=&\\frac{i^2}{\\left[(\\frac{p}{2}+k)^2-m^2\n+i\\epsilon\\right]\\left[(\\frac{p}{2}-k)^2-m^2+i\\epsilon\\right]}\n\\nonumber\\\\\n&\\times&\\int \\frac{d^4k'}{(2\\pi)^4}iK(k,k',p)\\Phi(k',p),\n\\end{eqnarray}\nwhere $p$ and $k$ are the total and relative four-momenta,\nrespectively, and $K$ is the interaction kernel. The bound state\nmass squared is $M^2=p^2$. For nonconfining interactions, the mass $M$\nis smaller than $2m$, allowing the introduction of the binding energy\n$B$ (defined positive) through the relation $M^2=(2m-B)^2$.\nIn the ladder approximation of the kernel, represented by the exchange\nof a scalar particle with mass $\\mu$, the kernel has the form\n\\begin{equation}\\label{ladder}\niK(k,k',p)=\\frac{i(-ig)^2}{(k-k')^2-\\mu^2+i\\epsilon},\n\\end{equation} \nleading to an attractive interaction and the possible emergence of\nbound states.\n\n\n\\subsection{General properties of the solutions} \\label{genprop}\n\nThe W-C model corresponds to the case\n$\\mu=0$ in Eq. (\\ref{ladder}). In the non-relativistic limit, this\nmodel leads to the well-known Coulomb bound state spectrum.\nCutkosky showed that \nin the relativistic case,\nthe BS amplitude,\nhenceforth limited to $S$-wave states, characterized by a principal\nquantum number $n=1,2,\\ldots$, can be represented in terms of $n$\nfunctions $\\left\\{ g_n^{\\nu} \\right\\}_{{\\nu}=0,1\\ldots n-1}$, depending on\na single scalar argument $z\\in[-1,+1]$, as \n\\begin{eqnarray}\\label{Phi}\n&& \\Phi_n(k,p)=- {i \\over \\sqrt{N_{tot}}}\n\\sum_{{\\nu}=0}^{n-1}\\int_{-1}^1g_{n}^{\\nu}(z)dz \\nonumber \\\\\n&& \\times\\frac{m^{2(n-{\\nu})+1}}\n{[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{2+n-{\\nu}}},\n\\ n=1,2,\\ldots\\ . \\nonumber \\\\\n&& \n\\end{eqnarray}\n$N_{tot}$ is a dimensionless normalization factor, deter\\-min\\-ed in\n\\ref{exprssff},\nensuring the condition $F_{el}(0)=1$ for the elastic form factor . \nThe factor $m^{2(n-{\\nu})+1}$ in the numerator is introduced to deal\nwith dimensionless $g_{n}^{\\nu}(z)$ functions.\n\nBy inserting (\\ref{Phi}) in the BS equation (\\ref{bs}), Cutkosky\nobtained [Eq. (14) of Ref. \\cite{cutk}] a system of homogeneous coupled\nintegral equations for the functions $g_n^{\\nu}$.\nFor S-waves, it reads\\footnote{A typo seems to exist in Eq. (14) of\nRef. \\cite{cutk}: the integration with respect to $t$ goes from $-1$\nto $+1$, and not from $0$ to $+1$, as can be verified from Eq. (13)\nof that reference. Notice that we use a slightly different notation, \nreplacing $k \\to \\nu$, with respect to the original work \\cite{cutk}.}:\n\\begin{small}\n\\begin{eqnarray}\\label{ECut}\n&&g_n^{\\nu}(z)= {\\lambda\\over2} \\sum_{{\\nu}'=0}^{\\nu} \\frac{ (n-{\\nu}+1)(n-{\\nu})}\n { (n-{\\nu}'+1) (n-{\\nu}')} \\int_{-1}^1dt \\int_0^1 dx\\nonumber \\\\\n &\\times& x(1-x)^{n-{\\nu}-1} \\int_{-1}^{+1}dz'\n \\frac{\\delta[ z -xt-(1-x)z'] }{ [1-\\eta^2(1-z^2) ]^{{\\nu}-{\\nu}'+1}}\n g_n^{{\\nu}'}(z'), \\nonumber\\\\\n&& \n\\end{eqnarray}\n\\end{small}\nwhere $\\lambda$ is related to the coupling constant $g^2$ of the\ninteraction kernel (\\ref{ladder}) by\n$\\lambda= {g^2\\over 16 \\pi^2m^2}, $\nand the total mass square $M^2$, eigenvalue of the system (\\ref{ECut}),\nappears through the parameter\n\\[ \\eta^2={M^2\\over 4m^2}. \\]\nIntegrating Eq. (\\ref{ECut}), first with respect to $t$ through\nthe $\\delta$ function, taking into account the bounds to be\nsatisfied by $t$ and $x$, and distinguishing the two cases, $z>z'$\nand $z'>z$, one obtains the set of equations\n\\begin{eqnarray}\\label{ECut2}\n g_n^{\\nu}(z)&=& {\\lambda\\over2} \\sum_{{\\nu}'=0}^{{\\nu}} c_n^{{\\nu}{\\nu}'} \\;\n \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-{\\nu}} \\over [Q(z')]^{{\\nu}-{\\nu}'+1}} \\;\n g_n^{{\\nu}'} (z'), \\nonumber \\\\\n && n=1,2,\\ldots , \\; \\; {\\nu}=0,1,...n-1, \n\\end{eqnarray}\nwhere we have introduced\n\\[ c_n^{{\\nu}{\\nu}'}=\\frac{ (n-{\\nu}+1)} { (n-{\\nu}'+1) (n-{\\nu}')}, \\]\n\\begin{equation}\\label{Q}\n Q(z)=1-\\eta^2(1-z^2), \n \\end{equation}\nand\n\\begin{equation}\\label{eq1bb}\nR(z,z')=\\left\\{\n\\begin{array}{ll}\n\\frac{1-z}{1-z'},& \\mbox{for $z'z$.}\n\\end{array}\n\\right.\n\\end{equation}\nBy expanding Eq. (\\ref{ECut2}), one is left with a $n\\times n$\ntriangular system of one-dimensional integral equations of the form\n\\begin{small}\n\\begin{eqnarray}\ng_n^0(z) & =& {\\lambda\\over2}\n \\left[ c_n^{00} \\int_{-1}^{+1} dz' \\; { [R(z,z')]^n \\over Q(z')}\n \\; g_n^0 (z') \\right], \\label{ECut2_0} \\\\\ng_n^1(z) &=& {\\lambda\\over2}\n \\left[ c_n^{10} \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-1} \\over [Q(z')]^2}\n \\; g_n^0 (z') \\right.\n \\nonumber \\\\\n &+& \\left.c_n^{11} \\int_{-1}^{+1} dz' \\;\n { [R(z,z')]^{n-1} \\over Q(z') } \\; g_n^1 (z') \\right], \n \\nonumber\\\\\ng_n^2(z) & =& {\\lambda\\over2}\n \\left[ c_n^{20} \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-2} \\over [Q(z')]^3}\n \\; g_n^0 (z') \\right.\n \\nonumber\\\\\n &+& \\left. c_n^{21} \\int_{-1}^{+1} dz' \\;\n {[R(z,z')]^{n-2} \\over [Q(z')]^2 } \\; g_n^1 (z')\\right.\n \\nonumber \\\\\n&+&\\left. c_n^{22} \\int_{-1}^{+1} dz' \\;\n { [R(z,z')]^{n-2} \\over Q(z') } \\; g_n^2 (z')\\right], \n \\nonumber\\\\ \n& & \\ldots \\ \\ \\ \\ \\ \\ \\ \\ldots \\ \\ \\ \\; \\ldots \\ \\ \\ \\ldots \\;\n \\ \\ \\ \\ldots\\ , \n \\nonumber\\\\\ng_n^{n-1}(z) &=& {\\lambda\\over2}\n \\left[ c_n^{n-1,0} \\int_{-1}^{+1} dz' \\; { R(z,z') \\over [Q(z')]^n}\n \\; g_n^0 (z') + \\ldots \\right. \n \\nonumber\\\\\n &+& \\left.c_n^{n-1,n-1}\n \\; \\int_{-1}^{+1} dz' \\; { R(z,z') \\over Q(z')} \\; g_n^{n-1}(z')\n \\right]. \\nonumber\n\\end{eqnarray}\n\\end{small}\n\nRemarkably, the function $g_n^0$, which allows the calculation of the\nenergy spectrum via the $M^2$-dependence of $Q$ [Eq. (\\ref{Q})], is\ntotally decoupled from the rest of the system.\nIt fulfills the single equation (\\ref{ECut2_0}), that we will hereafter\nwrite in terms of the fine structure coupling constant $\\alpha$, usual\nin the Coulomb problems:\n\\begin{eqnarray} \\label{gn}\n g_n^0(z)&=&\\frac{\\alpha}{2\\pi n} \\int_{-1}^1 {[R(z,z')]^n \\over Q(z')}\n \\; g_n^0 (z'), \n \\\\\n \\alpha&=&\\pi \\lambda = {g^2\\over 16\\pi m^2}.\n \\nonumber\n\\end{eqnarray}\n\nThe remaining equations allow the determination of $g_n^{k>0}$ -- and\nso of the BS amplitude (\\ref{Phi}) -- by solving an inhomogeneous\nproblem with an inhomogeneous term given by $g_n^0$.\nNotice that it is a quite unusual situation in Quantum Mechanics that\na part of the total system wave function, which, as we will see in what\nfollows is far from being dominant, \ndetermines the full spectrum of the system.\n\nAlthough the results presented here are limited to $S$-wave only, it is worth noticing that for $l\\neq 0$\nthe corresponding spherical function $Y_{lm}$ would appear as a prefactor\nin Eq. (\\ref{Phi}). \nThe angular momentum $l$ would enter in the system of equations (\\ref{ECut2}) and (\\ref{ECut2_0}), \nbut it turns out to be absent in the first equation (\\ref{ECut2_0}) determining the spectrum.\nAs a consequence, the BS amplitude would depend on $l$, while the spectrum would remain $l$- degenerate. \n\nIn view of its numerical solution, it is interesting to write Eq.\n(\\ref{gn}) in a differential form:\n\\begin{eqnarray}\\label{gndf}\n&&g_n^{0\\,\\prime\\prime}(z)+2(n-1)z(1-z^2)^{-1}g_n^{0\\,\\prime}(z)\n\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ -n(n-1)(1-z^2)^{-1}g_n^0(z)\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ +\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_n^0(z)=0,\n\\end{eqnarray}\nwith the boundary conditions $g_n^0(\\pm 1)=0$.\n\\par\n\nFor a fixed $n$, Eq. (\\ref{gndf}) has an infinite number of\nsolutions, labeled by an additional quantum number\n$\\kappa=0,1,2,\\ldots$, which also labels the corresponding discrete\nspectrum of mass squared eigenvalues $M_{n\\kappa}^{2}$. \nWe will use the notation $g^{\\nu}_{n\\kappa}$ to identify a particular\nsolution.\nThe function {$g_{n\\kappa}^0$} has $\\kappa$ nodes within the interval\n\\mbox{$]-1,+1[$} and a well-defined parity given by $\\kappa$ \\cite{cutk}:\n\\[g^0_{n\\kappa} (-z)= (-1)^{\\kappa} g^0_{n\\kappa}(z) \\]\nThe parity is also preserved inside the ensemble $\\{ g_{n\\kappa}^{\\nu} \\}$\nwhen varying $\\nu=0,1,\\ldots$ and this entails, through Eq. (\\ref{Phi}),\nthat for even (odd) values of $\\kappa$,\nthe BS amplitude $\\Phi(k,p)$ is an even (odd) function of the relative\nenergy $k_0^{}$ in the c.m. frame. \\par\n\nThe mass squared $M^2$ of the ground state $g_{10}^0$ as function of the\ncoupling constant $\\alpha$ is shown in Fig. \\ref{Fig_M2_alpha_10}.\nIts value vanishes for $\\alpha=2\\pi$. In the range $\\alpha \\in [0,2\\pi]$,\n$M^2\\ge 0$, the total mass of the system $M$ is well defined as well as\nits binding energy $B=2m-M>0$.\nThis determines the domain where this model is physically consistent\nwith a well-defined ground state.\nIt is worth mentioning, however, that the solutions of the BS equation,\nas well as its spectral parameter $M^2$, can be analytically continued\nfor $\\alpha>2\\pi$ \nwithout encountering any kind of singularity. This is illustrated with\nthe dashed line in the lower right corner of the figure. \nAll excited states lie above the $M^2(\\alpha)$ curve and thus can have\na well-defined $M$ even in the unphysical region.\n\n \\vspace{.5cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\mbox{\\epsfxsize=8.cm\\epsffile{M2_alphaext_1_0.eps}} \n\\hspace{.5cm}\n\\end{center}\n\\caption{(Color online) Dependence of the squared mass of the ground\n state ($n=1, \\kappa=0)$ on the coupling constant $\\alpha$. \n Beyond the critical value $\\alpha=2\\pi$, the solutions are smootly\n continued without any singularity but having negative values of\n $M^2$. \n The physical region, where the system has well-defined ground state\n mass and binding energy, is thus limited to $\\alpha\\in[0,2\\pi]$.}\n\\label{Fig_M2_alpha_10}\n\\end{figure}\n\nAmong the infinity of solutions existing for a given $n$, the one with\n$\\kappa=0$ coincides, in the limit \nof small binding energies $B\/m\\ll 1$, with the solution of the\nnon-relativistic Coulomb problem with main quantum number $n$. \nThis solution is called, following the original works of\nWick and Cutkosky \\cite{wick,cutk}, ``normal''.\nIndeed, these authors, analyzing in this limit the system of equations\nfor the functions $g_n^k(z)$ determining the BS amplitude (\\ref{Phi}), \nreproduced, for $\\kappa=0$, the Coulomb spectrum, i.e. the Balmer series \n\\begin{equation}\\label{Balmer}\nB_n=\\frac{m\\alpha^2}{4n^2}.\n \\end{equation}\nThis result corresponds to the Schr\\\"odinger equation with the potential\n$V(r)=-\\frac{\\alpha}{r}$. \nThe relativistic perturbative correction to\nthe binding energy (\\ref{Balmer}) was found in \\cite{FFT}; the binding\nenergy, incorporating it, reads\n\\begin{equation}\\label{B_Pert}\n B_n=\\frac{m\\alpha^2}{4n^2}\\left[ 1 - {4\\alpha\\over\\pi}\n \\ln\\left({1\\over\\alpha}\\right) + {\\mathcal O}(\\alpha) \\right]. \n \\end{equation}\n\nOn the contrary, the solutions corresponding to non-zero values of\n$\\kappa$ ($\\kappa=1,2,\\ldots$), have a spectrum totally decoupled\nfrom the non relativistic one.\nThey are genuinely of relativistic nature, without non-relativistic\ncounterparts, and were named \"abnormal\" by Wick.\n\nThese different behaviours are illustrated in Fig. \\ref{lambda_B} where\nwe have displayed the dependence of the coupling constant\n$\\lambda={\\alpha\\over\\pi}$ on the binding energy $B$\nfor the lowest solutions of the W-C model. \nUpper panel contains only the n=1 states with $\\kappa=0,1,2,3,4$. \nThe curve corresponding to $\\kappa=0$ (black solid line) is tangent\nto the non-relativistic (NR) one (black dashed line) from which\nit departures logarithmically, as it is visible, starting at\n$B\\approx 0.001$.\nThe perturbative results, provided by Eq. (\\ref{B_Pert}), are\nindistinguishable from the exact ones in the considered energy range.\nThose corresponding to $\\kappa>0$ (colored solid lines) do not have any\nnon-relativistic counterparts. \nLower panel represents the spectrum for $n\\ge 1$ states and different\nvalues of $\\kappa=0,1,2,3$.\nThe horizontal line $\\lambda=2$ ($\\alpha=2\\pi$) indicates the maximal\nvalue of the coupling constant ensuring a well-defined ground state. \n\n\\begin{figure}[h!]\n\\vspace{.5cm}\n\\begin{center}\n\\mbox{\\epsfxsize=8.2cm\\epsffile{lambda_B_mu_0.00_n_1_LogLog.eps}} \\\\\n\\vspace{1.cm}\n\\mbox{\\epsfxsize=8.cm\\epsffile{lambda_B_mu_0.00_n_LogLog_0.001_0.5.eps}} \n\\end{center}\n\\caption{(Color online) Spectrum of the W-C model as a function of the\n coupling constant $\\lambda(B)$ in the low energy limit. Upper panel\n corresponds to n=1 states with different values of $\\kappa=0,1,..$ .\n The case $\\kappa=0$ is compared\nto the non relativistic solution. Horizontal lines correspond respectively \nto the maximal values of the coupling constant for a well-defined\nground state ($\\lambda=2$) and to the minimal value for\nwhich the abnormal solutions exist ($\\lambda=1\/4$).\nLower panel contains the full spectrum for $n\\le6$ and $\\kappa\\le3$\nto make explicit the different crossings. Notice that the unphysical\nsolutions with odd $\\kappa$ (giving no contribution to the S-matrix)\nare naturally inserted in the spectrum.\n\\label{lambda_B}}\n\\end{figure}\n\n\nThe normal and abnormal solutions have also different domains of\nexistence with respect to the coupling constant $\\alpha$. \nAs a mathematical solution of the BS equation (\\ref{bs}), the normal\nsolutions exist for any (positive) values of $\\alpha$, although, as we\nhave already discussed,\nthey have a clear physical meaning only in the range $\\alpha\\in[0,2\\pi]$,\nwhere $M^2\\ge 0$.\nHowever, the very existence of the abnormal solutions (all of them)\nrequires a coupling constant greater than some critical value,\n$\\alpha\\ge{\\pi\\over4}$ ($\\lambda\\ge1\/4$). \nThis can be clearly seen from the results of Fig. \\ref{lambda_B}, where\nall the abnormal states (color line) were\nfound above the horizontal $\\lambda=1\/4$ line. \n\nWick and Cutkosky \\cite{wick,cutk} found the following approximate\nanalytic expression for the abnormal spectrum near the continuum\nthreshold ($B\\to 0$): \n\\begin{equation} \\label{abnspectr}\n M_{n\\kappa}^2\\simeq 4m^2\\left( 1 - e^{ -{(\\kappa-1)\\pi\/\n \\sqrt{{\\alpha\\over \\pi}-{1\\over 4}}}}\\right),\\;\\kappa=2,3,\\ldots\\ , \n\\end{equation}\nwhere the condition $\\alpha>\\pi\/4$ is explicitly obtained. \nAt $B\/m\\ll 1$ this spectrum vs. $\\kappa$ does not depend on $n$.\nIt also indicates that to be able to distinguish abnormal states from the\ncontinuum threshold on experimental grounds, the coupling constant\nshould be increased at least up to values of $\\alpha\\approx 4\\div 5$;\notherwise, for values of $\\alpha$ very close to $\\pi\/4$, the\nexponential in Eq. (\\ref{abnspectr}) is nearly zero and the discrete\nspectrum becomes hardly distinguishable from the continuum.\n\nIt is worth noting that the existence of a lower bound of the\ncoupling constant for the abnormal solutions is reminiscent of the\nmassive-exchange case, i.e. $\\mu\\ne0$ in the kernel (\\ref{ladder}),\nwhich was considered with some detail in \\cite{MC_PLB474_2000} both in\nthe BS and the Light-Front Dynamics frameworks.\nThe $B_{\\mu}(\\alpha)$ dependences (Figs. 5 and 7 of\n\\cite{MC_PLB474_2000}) are similar to those displayed in Fig.\n\\ref{lambda_B}), what suggest the possibility \nto associate a mass with the abnormal states.\nHowever, essential differences in the number of bound states for a\ngiven $\\mu$ remain: infinite in the W-C model and (at most) finite\nin the massive case.\n\nIn summary, the range of the coupling constants to be considered in\nthis model is $0<\\alpha<2\\pi$ ($0<\\lambda<2$) for the\nnormal states, i.e. $\\kappa=0$, and ${\\pi\\over 4}<\\alpha<2\\pi$\n($ 1\/4<\\lambda<2$) for the abnormal ones ($\\kappa>0$).\nOn another hand, according to Refs. \\cite{cia,nai}, the abnormal\nsolutions with odd\nvalues of $\\kappa$ do not contribute to the $S$-matrix and therefore only\nthose with even $\\kappa$ can have a physical meaning. \nIn the subsequent part of this work, we will concentrate on the latter case.\nFurthermore we will restrict ourselves, to the $l=0$ states with $n=1$\nand $n=2$.\n\nFor $n=1$ states, the sum (\\ref{Phi}) is reduced to a single term\ninvolving only the function $g_1^0$ satisfying the homogeneous integral\nequation\n\\begin{equation}\\label{g10_Int}\n g_1^0(z)=\\frac{\\alpha}{2\\pi}\\int_{-1}^1 \\frac{R(z,z')}{Q(z')} \\,\n g_1^{0}(z')dz',\n\\end{equation}\nor, equivalently, in its differential form\n\\begin{equation}\\label{g0}\n{g''}_1^0(z)+\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_1^0(z)=0,\n\\end{equation}\nwith the boundary conditions $g_1^0(\\pm 1)=0$.\nThe corresponding BS amplitude is expressed in terms of $g_1^0$ as\n\\begin{equation}\\label{Phi10}\n \\Phi_1(k,p)=\\int_{-1}^1\\frac{-im^3 \\,g_{1}^0(z)dz}{[m^2-\\frac{1}\n {4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{3}}.\n\\end{equation}\n\n\\bigskip\nFor $n=2$ states, the sum (\\ref{Phi}) involves two functions $g_2^0$\nand $g_2^1$. The function $g_2^0$ satisfies the homogeneous integral\nequation:\n\\begin{equation}\\label{g20_Int}\n g_2^0(z)=\\frac{\\alpha}{4\\pi}\\int_{-1}^1 \\frac{R^2(z,z')}{Q(z')} \\,\n g_2^{0}(z')dz',\n\\end{equation}\nand in differential form\n\\begin{eqnarray}\\label{eq1c}\n{g''_2}^0(z)&+&\\frac{2z}{(1-z^2)}{g'}_2^0(z)-\\frac{2}{(1-z^2)}g_2^0(z)\n\\nonumber\\\\\n&+&\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_2^0(z)=0,\n\\end{eqnarray}\nwith the boundary conditions $g_2^0(\\pm 1)=0$, while $g_2^1$ is determined\nfrom $g_2^0$ through the integral equation\n\\begin{eqnarray}\\label{g21}\n g_2^1(z)&=& \\frac{\\alpha}{6\\pi}\\int_{-1}^1\\frac{R(z,z') }\n {[Q(z')]^2}\n \\,g_2^{0}(z')dz' \n \\nonumber\\\\\n &+& \\frac{\\alpha}{2\\pi}\\int_{-1}^1\\frac{R(z,z')}\n {Q(z')} \\, g_2^{1}(z')dz',\n\\end{eqnarray}\nwhich can also be rewritten in the form of an inhomogeneous\ndifferential equation:\n\\begin{eqnarray}\\label{eq21c}\n&& {g''}_2^1(z)+\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_2^1(z)\n\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n= - \\frac{\\alpha}{3\\pi}\\frac{1}{(1-z^2)[Q(z)]^2}g_2^0(z).\n\\end{eqnarray}\n\\par\nThe BS amplitude (\\ref{Phi}) is now expressed in terms of two functions\n$g_2^1$ and $g_2^0$:\n\\begin{eqnarray}\\label{Phi2}\n \\Phi_2(k,p)&=&\\int_{-1}^1\\frac{-im^3 \\,g_{2}^1(z)dz}\n {[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{3}}\n \\nonumber\\\\\n &+&\\int_{-1}^1\\frac{-i m^5\\,g_{2}^0(z)dz}\n {[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{4}}.\n\\end{eqnarray}\n\n\n\\subsection{Numerical solutions for some selected states $g_{n\\kappa}^k$}\\label{g_num}\n\nWe present, in this subsection, the numerical results concerning the\nfirst states of the W-C spectrum. \nWe fix hereafter $m=1$ and the coupling constant to the value $\\alpha=5$.\nWe will consider along the work an ensemble of states with $n=1,2$ and\n$\\kappa=0,2,4$ that,\nfor the sake of simplicity in notation, will be numbered with No. 1-6\nin the Tables \\ref{tab1} and \\ref{tab2}.\n\n\\begin{table}[h!]\n\\begin{center}\n\\caption{Binding energy $B$ and two-body norm ($N_2$)\nof the low-lying normal ($\\kappa=0$) and abnormal\n($\\kappa=2$) states, for the coupling\nconstant value $\\alpha=5$.}\\label{tab1}\n\\begin{tabular}{cccll}\n\\hline\\noalign{\\smallskip} \nNo.& n & $\\kappa$ & $B$ & $N_2$ \\\\\n\\noalign{\\smallskip}\\hline \n\\noalign{\\smallskip}\n1&1&0 & 0.999259 &0.65 \\\\\n2&2&0 & 0.208410 & 0.61 \\\\\n3& 1 &2 &\n$3.51169 \\cdot 10^{-3}$\n& 0.094 \\\\\n4& 2 &2 &\n$1.12118 \\cdot 10^{-3}$\n& 0.077 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe binding energies for the lowest $n=1,2$ normal ($\\kappa=0$) and\nabnormal ($\\kappa=2$) states $g_{n\\kappa}^{\\nu}$ are presented in Table\n\\ref{tab1}.\nAll $\\nu=0$ components are arbitrarily normalized to $g_{n\\kappa}^0(0)=1$. \nThe corresponding solutions for the $n=1$ states -- $g_{10}^0$ and\n$g_{12}^0$ -- are displayed in Figs. \\ref{fig1} and \\ref{fig1p}.\nThey have comparable sizes\nand their nodal structure is determined by $\\kappa$ only.\n\n\\begin{figure}[h!]\n\\vspace{.8cm}\n\\begin{minipage}[h!]{8.7cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{g100_Nb1.eps}\n\\caption{$g_{10}^0$ for the normal state No. 1 of Table \\ref{tab1}.} \\label{fig1}\n\\end{center}\n\\end{minipage}\n\n\\begin{minipage}[h!]{8.7cm}\n\\vspace{.9cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{g120_Nb3.eps}\n\\caption{$g_{12}^0$ for the abnormal state No. 3 of Table \\ref{tab1}.}\\label{fig1p}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nThe two components $g_{2\\kappa}^0$ and $g_{2\\kappa}^1$ of the $n=2$\nstates are plotted in Fig. \\ref{g2} (state No. 2 with $\\kappa=0$)\nand Fig. \\ref{g2ab} (state No. 4 with $\\kappa=0$). \nThe component $\\nu=1$ is dominant in both cases, but for the state No. 4\nit is $\\sim 1000$ times larger (see the scaling factor in Fig. \\ref{g2ab}).\nThis enhancement is due to the $Q^2$ factor in the denominator of the\nright-hand-side of Eq. (\\ref{eq21c}), \nwhich, in the limit $B\\to 0$ and around $z=0$, behaves as\n$Q^2\\approx B^2 $.\nThus, for $n>1$ states with small binding energies, the component\n$g_{n\\kappa}^0$ that determines $M^2$ is\nnegligibly small with respect to the other components.\nWe will see, however, in the following section that they all\nplay equivalent roles in the construction of the BS amplitude itself \nand in the form factors.\n\n\\begin{figure}[h!]\n\\vspace{0.8cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5cm\\epsfbox{g200_g201_Nb2.eps}\n\\caption{Components $g_{20}^0$ and $g_{20}^1$ of the normal state No. 2 from Table \\ref{tab1}.}\\label{g2}\n\\end{center}\n\\end{figure}\n\\vspace{0.5cm}\n\n\\begin{figure}\n\\vspace{0.8cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5cm\\epsfbox{g220_g221_Nb4.eps}\n\\caption{$g_{22}^0$ and $g_{22}^1$ (scaled by a factor $10^3$) of the abnormal state No. 4 (Table \\ref{tab1}).}\n\\label{g2ab}\n\\end{center}\n\\end{figure}\n\n\\vspace{0.5 cm}\nIn the rightest column of Table \\ref{tab1} we have also included the\nnorm $N_2$ of the two-body contributions in the Fock space, as it is defined in \\ref{appN2}.\nFor the states with $n=1$, $N_2$ is given by Eq. \nof \\ref{appN2}\nand for $n=2$, by Eqs. ({\\ref{norm2}). We remark therein that the\ntwo-body norm $N_2$ for the abnormal ($\\kappa=2$) states is much smaller\nthan for the normal ($\\kappa=0$) ones.\nThis comparison concerns however states covering the two extreme cases\nin the spectrum: deeply bound states (Nos. 1 and 2) and\nnearthreshold ones (Nos. 3 and 4).\nTo better understand this difference, we have \nstudied the dependence of $N_2$ on the binding energy for the first\nnormal and abnormal states. Results are displayed in Fig. \\ref{N2_B}.\n\n\\begin{figure}[h!]\n\\vspace{0.9cm}\n\\begin{center}\n\\epsfxsize=7. cm\\epsfysize=5cm\\epsfbox{N2_B_N.eps}\\\\\\vspace{1.cm}\n\\epsfxsize=7. cm\\epsfysize=5cm\\epsfbox{N2_B_A.eps}\n\\caption{(Color online) $N_2$-dependence on the binding energy for\n the $n=1$ and $n=2$ states: normal (upper panel) and abnormal (lower\n panel).}\n\\label{N2_B}\n\\end{center}\n\\end{figure}\n\n\nThe upper panel concerns the normal states. \nThe behaviours of the the $n=1$ (black solid line) and $n=2$ (red\nsolid line) states are quite similar: \n$N_2$ decreases monotonically from $N_2\\approx 1$ when $B\\approx 0$\ndown to an asymptotic value when $B\\to 2m$. We found numerically\n$N_2(2m)\\approx 0.64$ for n=1 and $N_2(2m)\\approx 0.59$ for n=2.\nFor the ground state n=1, these limiting values were found analytically\nin \\cite{dshvk}, as well as the perturbative expansion\nin their vicinity. Thus, the limit $B\\to0$ is described by Eq.\n(\\ref{N2_1a}), i.e., \n\\[ N_2(B)= 1+{1\\over\\pi}\\sqrt{{4B\\over m}}\\ln\\left({4B\\over m}\\right). \\]\nAt $B\/m=10^{-5}$, this perturbative expansion gives $N_2=0.980$ in\nclose agreement with the black curve of Fig. \\ref{N2_B}.\nWe conclude from this study that the normal states are dominated by\ntwo-body norms. This is particularly true in the limit $B\\to 0$,\nwhere $N_2\\to 1$, but remains also true in all the energy domain,\nalthough decreasing with increasing $B$.\n\nA very different behaviour is observed with the abnormal states,\nrepresented in the lower panel of Fig. \\ref{N2_B}.\nAs one can see, the two body norm $N_2$ of these states not only remains\ncomparatively very small, but also vanishes in the non-relativistic\nlimit, making them, in this region, genuine many-body states.\nThe one order of magnitude observed in Table \\ref{tab1} for the\nbinding energies hides in fact\na deeper and striking difference between normal and abnormal BS states,\nindependent of their comparison with the non-relativistic spectrum.\nIt is provided by their two-body content: abnormal states do not have\nin the limit $B\\to 0$ any two-body contribution and have, thus,\ngenuine many-body structures.\nBeyond this limit the norm of the two-body sector remains extremely\nsmall. This is the reason why they are absent in the non-relativistic\nlimit reduced to the two-body Schr\\\"odinger equation.\n\n\nThe results presented in Table \\ref{tab1} are completed in\nTable \\ref{tab2} by studying the $\\kappa=4$ excitations of $n=1,2$ \nstates. The same conclusion holds, even in a more dramatic way.\nTheir two body norms are one order of magnitude smaller than for the\n$\\kappa=2$ states of Table \\ref{tab1}.\nThis can be expected due to their smaller binding energies and in view\nof the behaviour described in the lower panel of Fig. \\ref{N2_B}.\n\n\n\\begin{table}[h!]\n\\begin{center}\n \\caption{Same as in Table \\ref{tab1}, for the abnormal states with $\\kappa=4$.} \\label{tab2}\n\\begin{tabular}{cccll}\n\\hline\\noalign{\\smallskip} \nNo.& n & $\\kappa$ & $B$ & $N_2$ \\\\\n\\noalign{\\smallskip} \\hline\n\\noalign{\\smallskip}\n5&1 &4&\n$1.54091\\cdot 10^{-5}$ & $6.19\\cdot 10^{-3}$ \\\\\n6&2 &4&\n$4.95065 \\cdot 10^{-6}$ &\n$2.06\\cdot 10^{-5}$\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe abnormal solution $g_{14}^0$} for $n=1, \\kappa=4$ state (No. 5\nin Table \\ref{tab2}), is shown in Fig. ~\\ref{fig10_14}, displaying its\nmore involved nodal structure (4 zeros in $]-1,+1[$). \nThe functions {$g_{24}^{\\nu}$ of the abnormal state $n=2, \\kappa=4$\n(No. 6 in Table \\ref{tab2}), are plotted in Fig. \\ref{g240_g241_Nb6}.\nThe extreme smallness of the binding energy of this state\ngenerates a huge enhancement factors in the inhomogeneous equation\n(\\ref{eq21c}) (through the factor Q) which results in \na huge dominance of the $\\nu=1$ component in the full BS amplitude.\nNotice that $g_{24}^1$ has been reduced by a factor $10^5$ to become\ncomparable with $g_{24}^0$.\n\n\\begin{figure}[h!]\n\\vspace{1.2cm}\n\\begin{minipage}[h!]{8.5cm}\n\\begin{center}\n\\epsfxsize=8.cm\\epsfysize=5cm\\epsfbox{g140_Nb5.eps}\n\\caption{$g_{14}^0$ from state No. 5 in Table \\ref{tab2}.}\n\\label{fig10_14}\n\\end{center}\n\\end{minipage}\n\\vspace{1cm}\\\\\n\\begin{minipage}[h!]{8.5cm}\n\\begin{center}\n\\epsfxsize=8.cm\\epsfysize=5cm\\epsfbox{g240_g241_Nb6.eps}\n\\caption{(Color online) $g_{24}^{\\nu}$ from state No. 6 in Table \\ref{tab2}.}\n\\label{g240_g241_Nb6}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\\section{Electromagnetic form factors}\\label{FFs}\n\nWe suppose that one of the two constituent particles is charged. \nThe electromagnetic form factor of the system can be expressed in\nterms of its BS amplitude. It is enough to consider inelastic\ntransitions from an initial $| i\\rangle$ to a final $| f\\rangle$ state.\nThe elastic form factors are obtained from them as a particular case,\nwith $f=i$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics{triangle.eps}\n\\caption[*]{Feynman diagram for the electromagnetic form factor.\n\\label{triangle}}\n\\end{figure}\n\\par\nThe electromagnetic vertex $J_{\\mu}$, corresponding to a transition\n$| i\\rangle\\to | f\\rangle$ is shown graphically in Fig.~\\ref{triangle}. \nThe corresponding vertex amplitude reads (we use Itzykson and Zuber\n\\cite{IZ} conventions for the Feynman rules):\n\\begin{eqnarray}\\label{ffGam}\niJ_{\\mu}&=&\\int \\frac{d^4k}{(2\\pi)^4}\\,\n\\frac{i[-i(p+p'-2k)_{\\mu}]}{(k^2-m^2+i\\epsilon)}\n\\nonumber\\\\\n&\\times&\\frac{i\\left[-i\\overline{\\Gamma}\n\\left(\\frac{1}{2}p' -k,p'\\right)\\right]}\n{[(p'-k)^2-m^2+i\\epsilon]}\n\\frac{i\\left[i\\Gamma \\left(\\frac{1}{2}p-k,p\\right)\\right]}\n{[(p-k)^2-m^2+i\\epsilon]},\n\\end{eqnarray}\nwhere $\\Gamma(k,p)$ is the vertex function, related to\nthe BS amplitude by the equation\n\\begin{eqnarray}\\label{PhiG}\n& &\\Phi(k,p)=\\frac{\\Gamma(k,p)}\n{\\left[(\\frac{p}{2}+k)^2-m^2+i\\epsilon\\right]\n\\left[(\\frac{p}{2}-k)^2-m^2+i\\epsilon\\right]}.\\nonumber \\\\\n& &\n\\end{eqnarray}\n$\\overline{\\Gamma}$ is the conjugate of $\\Gamma$, obtained from the\nlatter by complex conjugation and use of the anti\\-chrono\\-logical\nproduct. A similar definition also holds for the BS amplitude\n$\\overline{\\Phi}$\n\\cite{nak69}.\n\\par\nThe electromagnetic vertex for the transition $i\\to f$ is expressed in\nterms of the BS amplitude as\n(see e.g. Eq. (7.1) in \\cite{cdkm}):\n\\begin{eqnarray}\\label{ffbs}\nJ_{\\mu}\n&=&i\\int \\frac{d^4k}{(2\\pi)^4}(p+p'-2k)_\\mu \\; (k^2-m^2)\n\\nonumber\\\\\n&\\times&\\overline{\\Phi}_f\\left(\\frac{1}{2}p'-k,p'\\right)\\Phi_i\n\\left(\\frac{1}{2}p-k,p\\right). \n\\end{eqnarray}\nIt has the following general decomposition in terms of two scalar\nfunctions (see\\footnote{We change the notations in comparison to Ref.\n\\cite{tff}, where the form factor $G$ was denoted $F'$.} \\cite{tff}):\n$F(Q^2)$ and $G(Q^2)$ \n\\begin{eqnarray}\\label{ffc}\nJ_{\\mu}&=&\\left[(p_{\\mu}+{p'}_{\\mu})+ ({p'}_{\\mu}-p_{\\mu})\\frac{Q_c^2}\n{Q^2}\\right]F(Q^2)\n\\nonumber\\\\\n&-& ({p'}_{\\mu}-p_{\\mu})\\frac{Q_c^2}{Q^2}G(Q^2).\n\\end{eqnarray}\nHere, $q=p'-p$, $Q^2=-q^2=-(p'-p)^2$ and\n\\begin{equation} \\label{Qc2}\nQ_c^2={M_f}^2-M_i^2,\n\\end{equation} \n$M_i$ and $M_f$ being the masses of the initial and final states,\nrespectively.\n\nSince $q\\makebox[0.08cm]{$\\cdot$} J=Q_c^2 G(Q^2)$, the above decomposition does not suppose,\nin general, current conservation $q\\makebox[0.08cm]{$\\cdot$} J=0$, which implies \n\\begin{equation}\\label{G}\nG(Q^2) \\equiv 0.\n\\end{equation} \nA direct proof of this result from the BS equation is presented in\n \\ref{proofG}.\nThe current conservation becomes a stringent self-consistency criterion\nof our results: the calculated form factor $G(Q^2)$ should\nbe identically zero, or very small within the numerical uncertainties.\n \nWe also note that since $J_{\\mu}$ [Eq. (\\ref{ffc})] is not singular\nat $Q^2=0$, the following relation should hold:\n\\begin{equation}\\label{FeqFp}\nF(0)=G(0). \n\\end{equation}\nEquation (\\ref{G}) then implies that $F(0)=0$ for the transition form\nfactors (for which $Q_c^2\\neq 0$).\n\\par\nFrom Eq. (\\ref{ffc}), the form factors are expressed throu\\-gh\n$J_{\\mu}$ as\n\\begin{eqnarray}\\label{FFp}\n& &F(Q^2) = \\frac{ (p+p')\\makebox[0.08cm]{$\\cdot$} J\\,Q^2 + q\\makebox[0.08cm]{$\\cdot$} J \\, Q_c^2}\n{[(M_f-M_i)^2 +Q^2][(M_f+M_i)^2+Q^2]},\\nonumber \\\\\n& &G(Q^2) = \\frac{q\\makebox[0.08cm]{$\\cdot$} J}{Q_c^2}.\n\\end{eqnarray}\nConsidering these formulas at $Q^2=0$ (which does not imply $q=0$),\none gets the relation\n$F(0)=\\left.\\frac{q\\makebox[0.08cm]{$\\cdot$} J}{Q_c^2}\\right|_{Q^2=0}=G(0)$, which\nreproduces Eq. (\\ref{FeqFp}). \n\\par\nThe expressions for the form factors are obtained by substituting\nin Eqs. (\\ref{FFp}) the current $J$ [Eq. (\\ref{ffbs})], then\nsubstituting the BS amplitudes (\\ref{Phi}), using the Feynman\nparametrization and integrating over $k$. In this way, we find the\nform factors in the form of integrals over products of functions\n$g_n^{\\nu}(z)$ and $g_{n'}^{\\nu'}(z')$. (Details of similar calculations can\nbe found in Ref. \\cite{ckm_ejpa}.) \nThe result for the transition form factor $F(Q^2)$ can be written\nin the form:\n\\begin{equation}\\label{ff}\nF(Q^2)=\\sum_{\\nu=0}^{n-1}\\sum_{\\nu'=0}^{n'-1}F_{nn'}^{\\nu\\nu'},\n\\end{equation}\nand similarly for $G(Q^2)$.\nThe expressions of the functions of the right-hand-side of this\nequation for the cases $n=n'=1$ and $n=n'=2$ are given in\n\\ref{exprssff}.\n\n\nIn the following subsections, we examine the numerical results for the\nelastic and transition form factors for some states from Tables\n\\ref{tab1} and \\ref{tab2}, corresponding to the coupling constant\n$\\alpha=5$. \n\n\n\\subsection{Elastic form factors $F_e(Q^2)$}\\label{el}\n\nThere are two regions of interest in studying the elastic form factors\n($F_e$):\nthe region near the origin, giving insight into the size of the system,\nand the asymptotic region $Q^2\\gg m^2$, related to the many-body structure\nof the wave function \\cite{matvmurtavk,brodsfarr,radyush}.\n\nWith the elastic form factor of a bound state, normalized to\n$F_e(Q^2=0)=1$, the squared radius is given by\n\\begin{equation}\\label{r2}\n= -6 \\left(dF_e\\over dQ^2\\right)_{Q^2=0} \n\\end{equation}\nand the root mean squared (r.m.s.) radius by $R=\\sqrt{}$. \nIn the non-relativistic theory, the size of a bound state scales, as\na function of its binding energy, as $R\\approx {1\\over\\sqrt{mB}}$. \n\nOn the other hand, as mentioned in the Introduction, the asymptotics\nof $F_e$ should qualitatively probe the compositeness of the state. \nAccording to \\cite{matvmurtavk,brodsfarr,radyush}, the elastic form\nfactors of a $n$-body system should decrease as $1\/(Q^2)^{n-1}$,\nwhere $n$ is the number of the constituents of the state.\nIt is, however, worth emphasizing here some essential differences of\nthe W-C model with the theoretical framework in which the above\nasymptotic behaviors have been obtained. The latter have been derived\nin theories characterized by dimensionless coupling constants, like\nQCD and the parton model. In the W-C model, bosonic fields interact\nby the exchange of a scalar particle; the coupling constant $g$ is\nthen dimensionful, having the dimension of mass.\nThis has an immediate consequence on the behavior of the BS\namplitude at large momenta. In the W-C model, as can be checked from\nEqs. (\\ref{bs}) and (\\ref{ladder}), the BS amplitude behaves at large\nmomenta as $1\/(k^2)^3$. In QCD, in the ladder approximation, where\nquarks interact by means of an exchange of a gluon field, the\nscalar part of the BS amplitude behaves at large momenta as\n$1\/(k^2)^2$, up to logarithms. Extending the comparison to bosonic\n$\\phi^4$-like theories, where the coupling constant is also dimensionless,\nthe interaction between two bosonic constituents is realized either\nby contact terms, or by the exchange of a two-particle loop; in both\ncases the large-momentum behavior of the BS amplitude is again \n$\\frac{1}{(k^2)^2}$, up to logarithms. The faster decrease of the\nBS amplitude in the W-C model affects the behaviors of the form factors: \none thus expects in this model behaviors of the type $1\/(Q^2)^n$, up\nto logarithms, instead of $1\/(Q^2)^{n-1}$. \n\nIn QCD, the number $n$ represents the number of valence quarks\n(including, eventually, the number of valence gluons, in the case\nof hybrids).\nThe Fock sectors of the state which contain the sea quarks, therefore\nwith higher $n$'s, are expected to decrease asymptotically faster and\nthus to display rapidly the asymptotic dominance of the valence quark\nsector, a phenomenon well observed on experimental grounds. In the\nW-C model, for the normal solutions, one indeed expects the dominance\nof the two-body sector, as was concluded in Sec. \\ref{WCsol}.\nHowever, for the abnormal solutions, the two-body sector is weakly\ncontributing to the composition of the corresponding states, which are\ndominated by higher sectors of the Fock space. One therefore expects\nhere a competition between the contributions of the higher sectors\nof the Fock space, which asymptotically decrease more rapidly, but\nhave large coefficients, and the contribution of the two-body sector,\nwhich dominates in the asymptotic region, but with a small coefficient.\n\nFor abnormal solutions, or hybrid states, of the \\mbox{W-C} model, a \nrefined analysis necessitates the distinction between three regimes in\nthe $Q^2$ evolution of the form factors, rather than two:\nthe very small $Q^2$ regime, determined by the binding energy (and \nequivalently by the rms radius), \nthe intermediate $Q^2$ region where the decrease is determined by\nthe many-body components (and therefore is fast)\nand the asymptotic region, where the many-body contribution is\nexhausted, and only the two-body contribution survives. \nSince in the W-C model the content of any state is a two-body component\nplus an indefinite number of exchange particles, \nthe asymptotic behavior of all the elastic form factors is finally\ndetermined by its two-body contribution. \nTherefore, the asymptotic $Q^2$-dependence should be the same, though\nwith different coefficients, for all elastic form factors. \nIn particular, we predict that the ratios of all elastic form factors\nshould tend to constants at $Q^2\\to\\infty$.\n\nWe first examine the $n=1$ states. They are defined by a single component\n$g_{1\\kappa}^0$, the same that determines the binding energy.\nWe have plotted, in Fig. \\ref{Fel_1_3}, in solid lines, the elastic\nform factors for the two $n=1$ states of Table \\ref{tab1}: on top,\nthe normal state No. 1 ($\\kappa=0$ , $B=0.999$), and at bottom, the\nabnormal state No. 3 ($\\kappa=2$, $B=0.00359$). \n\n\\vspace{0.5cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=7.5cm\\epsfysize=5cm\\epsfbox{FQ2_Nb1.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7.5cm\\epsfysize=5cm\\epsfbox{FQ2_Nb3.eps}\n\\end{center}\n\\caption{Elastic form factors of $n=1$ states of Table \\ref{tab1}\n(solid lines): No. 1 (normal) on top and No. 3 (abnormal) at bottom;\nthe dashed line corresponds to the state with $\\kappa=0$ with the same\nbinding energy (and rms radius) as the state No. 3 with\n$\\kappa=2$.}\n\\label{Fel_1_3}\n\\end{figure}\n\nThe corresponding rms radii are $R_1=1.16$ fm and $R_3=15.7$ fm,\nrespectively, which roughly scale as $\\sim{1\\over \\sqrt{mB}}$.\nBoth states have only one component $g_{n\\kappa}^0$, which in turn\ndetermines the energy. It is then natural that they have a similar\nbehaviour, close to that of the non-relativistic one.\nAt this level, one cannot see any drastic difference between a normal\nand an abnormal state. \nThe behaviours of their form factors are quite similar, however with\na much faster decrease\nfor the state No. 3, as expected from its smaller binding energy\nand its many-body structure. At $Q^2=1$, the value of $F_e$ for the\nstate No. 3 is three orders of magnitude smaller than that of the\nstate No. 1.\nIn order to disentangle the contributions of the binding\nenergy and of the asymptotic behaviour,\nwe have adjusted, at a second step, $\\alpha$ of the state No. 1 to\nhave the same binding energy as the state No. 3. \nThe result is displayed in dashed line on the lower panel: both curves\nare tangent to each other at the origin, implying that the rms radii\nare the same, but one notices that the abnormal state form factor still\ndecreases much faster than that of the normal one, by a factor 10 at\n$Q^2=1$. \n\n\\newcommand\\egal{\\mathop{=}}\nOne can show that the elastic form factors behave, when\n$Q^2\/m^2\\to \\infty$ as\n\\begin{eqnarray}\\label{Fe_asymp} \n F_e(Q^2)&=&\\Big({m^2\\over Q^2}\\Big)^2\n \\left[c_2 \\ln \\left({Q^2\\over m^2}\\right) + c_0 \\right] \\nonumber\\\\\n &+& {\\mathcal O}\\left(\\Big({m^2\\over Q^2}\\Big)^3\\ln \\left({Q^2\\over m^2}\\right),\n \\Big({m^2\\over Q^2}\\Big)^3\\right).\n\\end{eqnarray}\nFor the states $n=1$, the coefficient $c_2$ has a simple expression\nin terms of the BS amplitude:\n\\begin{equation} \\label{c_2n=1}\nc_2(n=1)=\\frac{1}{4\\pi^2}\\frac{[g_{1\\kappa}^{0\\,\\prime}(-1)]^2}{N_{tot}},\n\\end{equation}\nwhere $g'(-1)$ is the derivative of $g$ with respect to $z$ at $z=-1$\nand $N_{tot}$ is the normalization factor that ensures the condition\n$F_e(0)=1$ when $g$ is arbitrarily normalized (its expression is given\nin Eq. (\\ref{Ntot_1})). The expression of $c_0$ is more complicated and\ndepends on the bound state mass $M$, as well as on the function\n$g$ over the whole region of $z$ in the interval $[-1,+1]$.\n\nFor the normal state $n=1,\\kappa=0$, $c_0$ is negative in general,\nbut changes sign and becomes positive for small binding energies.\nThe expression of $g$ takes a simple\nform in the two extreme cases of non-relativistic limit and maximal\nbinding energy ($M=0$). Normalizing $g$ so that $g(0)=1$,\none has in the first case $g(z)=(1-|z|)$, with\n$N_{tot}=1\/(32\\pi\\alpha^5)$, and in the second case $g(z)=(1-z^2)$,\nwith $N_{tot}=1\/(270\\pi^2)$ \\cite{wick,cutk}. One obtains, in these\ntwo extreme cases, the asymptotic behaviors \\cite{dshvk}\n\\begin{eqnarray}\n\\label{ff_NR} \nF_e(Q^2)&\\simeq& \\frac{8\\alpha^5}{\\pi}\\Big({m^2\\over Q^2}\\Big)^2\n\\left[\\ln \\left({Q^2\\over m^2}\\right) + \\frac{2\\pi}{\\alpha} \\right]\n\\nonumber\\\\\n& & (\\mathrm{non-relativistic\\ limit}\\; M\\to 2m), \\\\ \n\\label{ff_M0}\nF_e(Q^2)&\\simeq& 270 \\Big({m^2\\over Q^2}\\Big)^2\n\\left[\\ln \\left({Q^2\\over m^2}\\right)- 4\\right] \\nonumber \\\\ \n& & (\\mathrm{ultra-relativistic\\ case}\\; M=0).\n\\end{eqnarray}\nNotice that in Eq. (\\ref{ff_NR}), we have neglected\n$\\alpha$-indepen\\-dent\nconstant factors in front of the additive term $2\\pi\/\\alpha$.\nThe fact that $c_0$ is generally negative, except for small binding\nenergies, where it can, however, take a large value (proportional to\n$1\/\\alpha$), has as a main consequence the screening of the logarithmic\ntail, requiring, for a numerical analysis of the asymptotic\nbehaviors, very large values of $Q^2$ ($Q^2\\gg 100\\div 1000\\ m^2$). \n\nIn order to put in evidence the asymtptotic behavior (\\ref{Fe_asymp})\nand to determine its leading terms we have computed the\n``reduced form factor'' $\\bar{F}_e(Q^2)$, defined as \n\\begin{eqnarray}\\label{Coefs_asymp} \n\\bar{F}_e(Q^2) &\\equiv& {1 \\over \\ln\\left({Q^2\\over m^2}\\right) } \\;\n\\Big(\\frac{Q^2}{m^2}\\Big)^2 \\; F_e(Q^2) \\nonumber\\\\\n&_{\\stackrel{{\\displaystyle=}}\n{Q^2\\to\\infty}}&\nc_2 + {c_0 \\over \\ln\\left({Q^2\\over m^2}\\right)}, \n\\end{eqnarray}\nwhich should tend, when $Q^2\\to \\infty$, to a\nconstant (up to logarithmic corrections).\nThe asymptotic coefficients $c_i$ can be extracted from $\\bar{F}_e(Q^2)$\nand its derivative at a given $Q^2$ with the relations\n\\begin{equation}\\label{c_0}\n c_0= - \\; { d\\bar{F}(Q^2)\\over dQ^2} \\; Q^2\n \\ln^2\\left({Q^2\\over m^2}\\right)\n\\end{equation}\nand \n\\begin{equation}\\label{c_2}\nc_2= \u00a0\\bar{F}(Q^2) - {c_0 \\over \\ln\\left({Q^2\\over m^2}\\right)} \n\\end{equation}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\vspace{0.7cm}\n\\epsfxsize=7.5cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb1_1000.eps}\n\\vspace{0.8cm}\\\\\n\\epsfxsize=7.5cm\\epsfysize=5.5cm\\epsfbox{c0_c2_Nb1_1000_NON.eps}\n\\end{center}\n\\caption{Asymptotic behaviours of the elastic form factor of the $n=1$\nstate No. 1 of Table \\ref{tab1}. Upper panel: the elastic form factor\n$F_e$ multiplied by $Q^{\\sigma}$ with $\\sigma=2$ (red), $\\sigma=4$ (blue), and $\\sigma=4$\ndivided by $\\ln(Q^2)$. \nLower panel: the asymptotic coefficients $c_0$ and $c_2$\ndefined in Eq. (\\ref{Coefs_asymp}).}\\label{QnuFQ2_1}\n\\end{figure}\n\n\nThe results for the $n=1$ state No. 1 from Table \\ref{tab1} are\ndisplayed in Fig. \\ref{QnuFQ2_1}.\nThe upper panel represents the elastic form factor multiplied by\n$Q^{\\sigma}$ with $\\sigma=2$ (red line), $\\sigma=4$ (blue line)\nand $\\sigma=4$ divided by the logarithmic term, as in Eq.\n(\\ref{Coefs_asymp}), to exhibit the asymptotic behaviour derived in\nEq. (\\ref{Fe_asymp}) \nand the important contribution that the logarithmic term can have\nin the domain ${Q^2}\\in[0,1000]$, even at $Q^2 \\sim 1000$. The\nlatter is seen in the difference between the blue and black lines,\nwhich exactly correspond to $\\bar{F}_e$.\nIn the lower panel we have plotted the coefficients $c_0$ and $c_2$\nin the ``asymptotic'' domain ${Q^2}\\in[100,1000]$, together\nwith the full reduced form factor $\\bar{F}_e$. They already show a nice\nconvergence at ${Q^2}=1000$, but the difference between $\\bar{F}_e$\nand its asymptotic value $c_2$ remains sizeable, due to the large\ncontribution of $c_0$, which decreases very slowly. We have also\nchecked the stability of our results with respect to the number\nof grid points ($n_{grid}=400,800,1600$) used in computing the form factors:\nthe sensitivity is not visible by eyes and is not significant in our analysis.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\vspace{0.5cm}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{QnuFQ2_Nb3.eps}\\vspace{0.8cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{c0_c2_Nb3_100.eps}\n\\end{center}\n\\caption{Same as in Fig. \\ref{QnuFQ2_1}, but for the $n=1$ abnormal\nstate No. 3 from Table \\ref{tab1} .}\\label{QnuFQ2_3}\n\\end{figure}\n\nFig. \\ref{QnuFQ2_3} contains the same results for the $n=1$ abnormal\nstate No. 3.\nDue to the faster decrease of the corresponding elastic form factor,\n(see the lower panel of Fig. \\ref{Fel_1_3}) the asymptotic regime is\nreached at $Q^2\\approx 50$, with the asymptotic constant\n$c_2\\approx 0.0004$, which is\nseven orders of magnitude smaller than for the normal state No. 1.\n\n\n\nFigures \\ref{Fel_2} and \\ref{Fel_4} contain the elastic form factors\nof the two $n=2$ states.\nThe result for the normal state No. 2 ($\\kappa=0, B=0.2084$) is\nrepresented in the upper panel of Fig. \\ref{Fel_2}.\nOne first observes a much faster decrease than for the $n=1$ state No. 1\n(upper panel of Fig. \\ref{Fel_1_3}), having comparable binding energies:\none order of magnitude at $Q^2 =1$.\nOne also remarks the appearance of two zeroes in the form factor, which\nbecomes negatives in the range $Q^2\\in[1.5,3.0]$. This is\na consequence of the complex structure of the state.\nIndeed, the different contributions $F^{\\nu\\nu'}$ depending on\n$g_{20}^\\nu g_{20}^{\\nu'}$ are indicated in the lower panel.\nAs one can see, the physical $F_e$ results from strong cancellations\nof terms which have opposite signs and are one order of magnitude larger\nthan the physical value they build. \nAlthough these components are of the same order, the contribution due\nto $g^0_{20}$ -- which determines the binding energy of the state --\nis far from being dominant.\n\n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{FQ2_Nb2_20.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{FQ2_Nb2_Dec.eps}\n\\caption{Upper panel: elastic form factor of the normal state No. 2 of\nTable \\ref{tab1} ($n=2$, $\\kappa=0$). The different contributions\n$F^{\\nu\\nu'}$ depending on $g_{20}^\\nu g_{20}^{\\nu'}$ are indicated in the lower\npanel.}\n\\label{Fel_2}\n\\end{center}\n\\vspace{0.5cm}\n\\end{figure}\n\nThe elastic form factor of the abnormal state No. 4\n($\\kappa=2, B=0.00112$) is displayed in Fig. \\ref{Fel_4}.\nThe same remark concerning the faster decrease than the n=1 state\nNo. 3 with comparable binding energy holds.\nNotice also the non trivial structure -- similar to a diffraction\npattern -- seen at $Q^2\\approx 0.01$ and detailed in the lower panel.\nSuch a structure, as well as the zeroes in upper panel of Fig \\ref{Fel_2},\nis totally unusual in a two-scalar system interacting by the simple\nkernel (\\ref{ladder}) and indicates the complexity of the wave function\nfor any state solution with $n>1$, be it normal or abnormal.\nThe decomposition of $F_e$ in terms of the different components\n$F^{\\nu\\nu'}$ is similar than for the state No. 2, i.e., strong\ncancellations occur among opposite sign larger terms. \n \n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5.5cm\\epsfbox{FQ2_Nb4.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7cm\\epsfysize=5.5cm\\epsfbox{FQ2_Nb4_Zoom2.eps}\n\\caption{Elastic form factor of the abnormal state No. 4 of Table\n\\ref{tab1} ($n=2$, $\\kappa=2$). The non-trivial structure at\n$Q^2\\approx 0.01$ is detailed in the lower panel.}\\label{Fel_4}\n\\end{center}\n\\end{figure}\n\nThe corresponding rms radii, extracted using Eq. (\\ref{r2}), are\n$R_2=3.8$ fm (state No. 2) and $R_4=49.0$ fm (state No.4).\nJust on the basis of their binding energies one should expect\ntwice smaller values.\nThe reason is again the complex structure of the BS amplitude of\n$n>1$ states, with a $\\nu>0$ dominating component (by a factor of\n$10^3$ in state No. 4) that plays no role in determining the\nbinding energy -- and so for the spatial extension -- of the system.\n\n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb2_1000.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb4_1000.eps}\n\\end{center}\n\\caption{Asymptotic behaviour of the elastic form factors of $n=2$\nstates of Table \\ref{tab1} (solid lines): No. 2 (normal) on top and No. 4 (abnormal) at bottom.}\\label{QnuFQ2_24}\n\\end{figure}\n\nAs it was the case for the $n=1$ states, the comparison of the elastic\nform factors of the $n=2$ normal and abnormal\nstates (upper panels of Figs. \\ref{Fel_2} and \\ref{Fel_4}) shows that\nthe abnormal state form factors decrease faster than the normal ones\nas functions of $Q^2$.\nAt $Q^2=1$ the ratio normal\/abnormal is three orders of magnitude.\nEven after adjusting the coupling constant of state No. 2, to have\nthe same binding energy than the state No. 4, the conclusion remains\nunchanged.\n\nIt is also interesting to examine the asymptotic behaviour, which is\nsupposed to have the same form (\\ref{Coefs_asymp}) than for $n=1$ states.\nThis is done in the two panels of Figure \\ref{QnuFQ2_24}. \nThey show again the importance of logarithmic corrections and the\ndifferent orders of magnitudes of the asymptotic constant $c_2$ between\nnormal and abnormal sates. Notice the scaling factor introduced in some\nof the plots to include the comparison in the same frame.\n\nThe ratios of form factors for the normal states Nos. 1 and 2 and for\nthe abnormal ones 3 and 4 are shown in Fig. \\ref{rat1234} (upper panel).\nThey indeed tend to constants. The value of this constant is $\\sim 7$.\nThe ratio of form factors for the abnormal states No. 3 and the normal\none No. 1 is shown in the lower panel of the same figure. It also tends\nto a constant. The value of this constant is $\\sim 10^{-5}$. \nNote that the ratio of form factors of different nature (abnormal\/normal)\nis much smaller than normal\/normal and abnormal\/abnormal, as expected.\nSurprisingly, the ratios normal\/normal and abnormal\/abnormal are the same.\nThese asymptotic behaviors of the elastic form factors bring additional\narguments in favor of the interpretation of the abnormal states of the\nW-C model as hybrids. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5cm \\epsfbox{rat_12_34.eps} \\vspace{0.5cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{rat_31.eps} \n\\caption{ (Color online) \nUpper panel: Dotted curve is the ratio of the form factors $F(Q^2)$ for\nthe normal ground state No. 1 and for the normal excited state No. 2. \nSolid curve is the same for the abnormal states No. 3 and No. 4.\nLower panel: The ratio of the form factors $F(Q^2)$ for the abnormal\nstate No. 3 from Table 1 and the normal one No. 1.}\\label{rat1234}\n\\end{center}\n\\end{figure}\n \n\\subsection{Transition form factors $F_{if}(Q^2)$}\\label{trF}\n\nFor the sake of completness in the study of abnormal solutions of the\nW-C model we present here the results for the transition form factors.\nThere are four states in Table \\ref{tab1} and, hence, six possible\ntransitions between them. \nThe corresponding transition form factors $F$ are shown in Figs.\n\\ref{Ftr1_234} and \\ref{F34_23_24}.\nThe comparison reveals a hierarchy of the transition form factors.\nIn Fig. \\ref{Ftr1_234} we can see that the form factor for the\ntransition between two normal states, No. 1\n($n=1,\\kappa=0$) $\\to$ No. 2 ($n=2,\\kappa=0)$ (upper panel), dominates, \nby a factor $\\sim 100$, over the maximal values of the normal $\\to$\nabnormal transitions (central and lower panels).\n\nThe transition form factor $F(Q^2)$ between two abnormal states, No. 3\n($n=1,\\kappa=2$) $\\to$ No. 4 ($n=2,\\kappa=2$), is displayed in Fig.\n\\ref{F34_23_24} (upper panel). Its maximal value has the same\norder of magnitude as the normal-normal one, \nthough it decreases much faster. At last, the form factors for the\ntransitions between the normal and abnormal states \n(Figs. \\ref{Ftr1_234} and \\ref{F34_23_24}, both central and bottom\npanels) are approximately 100 times smaller than the\nabnormal$\\leftrightarrow$abnormal form factor. This hierarchy is\napparently related to the need of rebuilding the state structure for\nthe normal$\\leftrightarrow$abnormal transitions.\n\nAt first glance, this dominance can be simply due to the very different\nbinding energies: two such states will have a small overlap,\nwithout invocating any abnormal character. Indeed, one can hardly\nseparate unambiguously the effect of different structures from the\ndifferent binding energies (the latter result in different wave functions).\nThat is why we have carried out the complex analysis based on the behavior\nof the form factor (elastic and inelastic) and on the content of the\nFock sectors.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{Ftr12.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{Ftr13.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{F14.eps}\n\\caption{Transition form factors from normal state No. 1\n($n=1,\\kappa=0$) to other states listed in Table \\ref{tab1}.\n$1\\to2$: No.2 ($n=2, \\kappa=0$, normal) (upper panel);\n$1\\to3$: No. 3 ($n=1, \\kappa=2$, abnormal) (central panel) and\n $1\\to4$: No.4 ($n=2,\\kappa=2$, abnormal) (lower panel).}\n\\label{Ftr1_234}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{F34.eps} \\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{Ftr20_12.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{Ftr24a.eps}\n\\caption{Same as in Fig. \\ref{Ftr1_234} between \n$3\\to 4$: No. 3($n=1,\\kappa=2$, abnormal) and No. 4 ($n=2,\\kappa=2$,\nabnormal)\n(upper panel); $2\\to 3$: the state No. 2 ($n=2,\\kappa=0$, normal) and\nthe No. 3 ($n=1,\\kappa=2$, abnormal) (central panel) and\n$2\\to 4$: the state No. 2 ($n=2, \\kappa=0$, normal) and \nNo. 4 ($n=2,\\kappa=2$, abnormal) (lower panel).}\\label{F34_23_24}\n\\end{center}\n\\end{figure}\n\nFor all the transitions that were considered in this section we have\ncalculated simultaneously the transition form factor $G(Q^2)$.\nThe contraction of the electromagnetic current $J$ with the momentum\ntransfer $q$ results in $q\\makebox[0.08cm]{$\\cdot$} J=Q_c^2G(Q^2)$ [Eq. (\\ref{FFp})].\nTherefore, the current conservation implies $G(Q^2)\\equiv 0$ for any $Q^2$\n[Eq. (\\ref{G})]. A formal proof of this equality, using the BS\nequation, is given in \\ref{proofG}.\nComputing a quantity that we know from the first principles (current\nconservation) that should be identically zero could be in principle\nconsidered, at most, as being superfluous.\nHowever, as is seen from the derivation given in \\ref{proofG}, \nthis property is directly related to the fact that $g$ is\nindeed a solution of the BS equation\nand the form factors, mainly the 3D integrals (\\ref{Fe_kkp}) and\n(\\ref{Fif_kkp}), have been accurately computed.\nIt thus constitutes a test for our numerical solutions.\nSimilar tests were successfully carried out in Ref. \\cite{tff} for the\nform factors corresponding to the electro-desintegration of a bound\nsystem (the transition discrete $\\to$ continuous spectrum).\n\nThe kind of results obtained in computing $G(Q^2)$ is illustrated in Figs.\n\\ref{G_23}, in a single example corresponding to the transition between \nNo. 2 $(n=2,\\kappa=0)$ $\\to$ No. 3 $(n=1,\\kappa=2)$ states.\nAs in the elastic case, the form factor $G(Q^2)$ results from the sum\nof two terms: $G^{10}(Q^2)$ (proportional to $g_2^1g_1^0$ ) and\n$G^{00}(Q^2)$ (proportional to $g_2^0g_1^0$). They are indicated\nrespectively at upper panel in dashed ($G^{10}(Q^2)$) and \nin dotted ($G^{00}(Q^2)$) lines. The sum of them, i.e., the full form\nfactor $G(Q^2)$, is indicated by a thick solid line and is\nindistinguishable from zero at the scale of the figure.\nIt is in fact $\\approx 10^{-6}$.\n\nNote however that the sensitivity of $G(Q^2)$ to the accuracy in solving\nthe BS equation, that is, in computing the functions $g(z)$ and in\ncalculating the 3D integrals (\\ref{Fif_kkp}), is very high. \nThe result $G(Q^2)\\equiv 0$ is due to delicate cancellations\nbetween terms which are several orders of magnitude greater.\nA small error in these calculations results in non-zero $G(Q^2)$. This\nis demonstrated in the lower panel of Fig. \\ref{G_23}.\nThus, an error in computing the binding energy, e.g., \nsetting $B_i= 3.512 \\cdot 10^{-3}$ instead of $B_i= 3.51169 \\cdot 10^{-3}$\nand $B_f=0.2084$ instead of $B_f = 0.2084099$, provides\n$G(Q^2) \\approx -0.005$.\nThis value is comparable with the maximum value of the corresponding\nform factor $F(Q^2)$ (in central panel of Fig. \\ref{F34_23_24}).\nAn apparent violation of current conservation would hide in fact a lack\nof accuracy in the computational procedure.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=6.5cm\\epsfysize=5cm\\epsfbox{Fp23b.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=6.5cm\\epsfysize=5cm\\epsfbox{Fp23aa.eps}\n\\caption{(Color online) Contributions to the $2\\to 3$ transition\nform factor $G(Q^2)$: \nthe dashed line is $G^{10}(Q^2)$ and the dotted line $G^{00}(Q^2)$.\nThe sum of them (solid line) is the full form factor $G(Q^2)$\n(upper panel):\nit is of the order of $10^{-6}$ and results from contributions of\nseveral orders of magnitude greater.\nA small error in computing the binding energy, e.g., \n$B_i= 3.512 \\cdot 10^{-3}$ instead of $B_i= 3.51169 \\cdot 10^{-3}$\nand $B_f=0.2084$ instead of $B_f = 0.2084099$, would provide\n$G(Q^2) \\approx -0.005$ (lower panel), a size comparable with $F(Q^2)$ in\nFig. \\ref{F34_23_24}.}\\label{G_23}\n\\end{center}\n\\end{figure}\n\nTo summarize this section, the comparison of the elastic form factors\npresented in Figs. \\ref{Fel_1_3}, \\ref{Fel_2}, \\ref{Fel_4},\nof the normal states with the abnormal ones,\nshows that the elastic form factors of the abnormal states vs.\n$Q^2$ decrease much faster than for the normal ones.\nAt $Q^2\\sim 1$, the abnormal form factors are about $10^3$ times\nsmaller than the normal ones, whereas the behaviors of the elastic\nform factors of the normal states with $n=1$ and $n=2$ remain very\nclose to each other. These observations confirm that the abnormal\nstates are dominated by the many-body Fock states\n\\cite{matvmurtavk,brodsfarr,radyush}. \n\\par\nThe transitions between the normal and abnormal states, in comparison\nto the normal-normal and abnor\\-mal-abnormal transitions, are also\nsuppressed. This suppression indicates that the normal and abnormal\nstates have different structures and the transitions between them\nrequire the rebuilding of the states.\n\\par\nThe quality of our numerical calculation is quite sufficient to\njustify the above conclusions. This is demonstrated in \nFig. \\ref{G_23} for the transition form\nfactor $G(Q^2)$. As is explained in Sec. \\ref{FFs} and proved in\n \\ref{proofG}, the electromagnetic current conservation\nrequires $G(Q^2)=0$. This is indeed observed in Fig. \\ref{G_23}, top,\n(solid line) as a result of rather delicate\ncancellations of several contributions. \nNumerical changes of $B$ and $g(z)$, which seem insignificant,\nmay noticeably change the value of $G(Q^2)$ (Fig. \\ref{G_23}, bottom). \n\nThe calculations of the form factors for the transitions to the states\ngiven in the Table \\ref{tab2}, with the precision used so far, are\nunstable and require much higher precision. We do not present them here.\n\n\n\\section{Concluding remarks}\\label{concl}\n\nOur present analysis shows that the abnormal solutions\nof the W-C model have a different internal structure than\nthe normal ones, which can be traced back to their decomposition\nproperties into Fock space sectors on light-front planes. \nThis constitutes a genuine property of these states and we propose it as\nan alternative characteristic to the traditional explanation in terms of\ntemporal degrees of freedom excitations.\nWhereas the normal solutions are dominated by the two-body Fock sector\nmade of the two massive\nconstituents, the abnormal ones are dominated by the Fock sectors made of\nthe two massive constituents and several or many massless exchange\nparticles. This feature is also manifested through the fast decrease\nof the electromagnetic form factors of the abnormal states, signalling\ntheir many-body compositeness. Therefore, the abnormal states do\nnot appear as pathological solutions of the BS equation, but rather as\nsolutions having specifically a relativistic origin, through the\ndominance, in their internal structure, of the massless exchange\nparticles. \n\nAnother particular feature of the abnormal solutions is the\nrelatively large value of the coupling constant needed for their\nexistence ($\\alpha>\\pi\/4$). While the stability condition of the\nW-C model also requires that $\\alpha$ be bounded by the\nupper value $2\\pi$, the corresponding window of permissible values\ndoes not belong to the domain of perturbation theory and\nthe question of the validity of the ladder approximation can be raised.\nThis question has been examined in Ref. \\cite{alkofer} in the light\nof the incorporation into the model of the renormalization effects.\nIt turns out that the above\ndomain of values of the coupling constant is incompatible with\na consistent treatment of such effects. The renormalization constants \nviolate the inequalities which follow from the positivity conditions\ncoming from the spectral functions of the renormalized fields.\n\nThe latter result brings us to questioning the effect of the\nhigher-order multiparticle exchange diagrams. This problem has been\ndealt with in Ref. \\cite{jalsazdj} in a model of QED, where\nthe two massive constituents are static and tied at fixed positions in\nthree-dimensional space.\nThe abnormal solutions corresponding essentially to excitations of\ndegrees of freedom described by\nthe relative time variable (or equivalently, of the relative energy\nvariable), this model should rather preserve their possible existence.\nIt turns out that in this configuration, the two-particle Green's\nfunction is exactly calculable: it does not display any abnormal\ntype of bound state in its structure; only the normal ground-state\nis present in the spectrum. On the other hand, the BS equation, in\nthe ladder approximation, still continues exhibiting abnormal\nsolutions. A similar conclusion is also obtained\nfrom a different approach, based on a three-dimensional reduction\nof the BS equation with the inclusion of multiparticle exchange\ndiagrams \\cite{bijteb}.\n\nThe above considerations \nsupport, as mentioned, another\ninterpretation of the abnormal solutions of the W-C model.\nIt is possible that excitations of the degrees of freedom described \nby the relative time variable correspond, from the point of view of\nthe Fock decomposition, to filling of the higher Fock sectors.\nThese two interpretations may not contradict each other, but rather\nbe compatible.\n\n\nIn the W-C model, the even-relative-ener\\-gy abnormal\nsolutions appear as theo\\-retically acceptable states, {and are a\nconstitutive part of the corresponding S-matrix. The fact that\nthey are dominated in Fock space by the many-body massless exchange\nparticles may suggest that they are a kind of ``hybrid'' states.\nThey might represent the Abelian scalar analogs of the hybrids that\nare searched for in QCD, which are coupled essentially to a pair of\nquark-antiquark and one or several gluon fields. Here, however,\nthe non-Abelian property of the gauge group, as well as the existence\nof gluon self-interactions, make the latter states better adapted\nfor experimental, as well as theoretical, investigations. On the\nother hand, the possible relevance of the multiparticle exchange\ndiagrams in the kernel of the BS equation remains a key ingredient\nfor the ultimate conclusion as to whether the W-C model\nmay have any experimental impact. Note, however, that it is natural\nto expect that since the multiparticle exchanges add extra exchange\nparticles in the intermediate states, \nthey do not reduce but rather increase the higher Fock components. \n\nIn any event, in spite of the fact that the W-C model\nis an oversimplified model, it nevertheless contains the phenomenon\nof the particle creation and gives an interesting example of natural\ngeneration of hybrid states.\nTherefore, the hybrid systems can naturally exist in more sophisticated\nfield theories and be detected in appropriate experiments.\n\nThe question of a possible existence of abnormal states in the case\nof massive-particle exchanges is currently under investigation by\nthe present authors.\n\n\\par\n\\bigskip\n\n\\begin{acknowledgements}\nV.A.K. is grateful to the theory group of the Institut de Physique\nNucl\\'eaire d'Orsay (IPNO) for the kind hospitality and financial\nsupport during his visits.\nH.S. acknowledges support from the EU research and innovation program\nHorizon 2020, under Grant Agreement No. 824093.\n\\end{acknowledgements}\n\\par\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nHom-Lie algebras are a generalization of Lie algebras, formalizing an algebraic structure which appeared first in quantum deformations of Witt and Virasoro algebras, see for example \\cite{AizawaSaito,ChaiIsKuLuk,CurtrZachos1,Kassel1,Hu}. A quantum deformation or a $q$-deformation of an algebra of vector fields is obtained when replacing a usual derivation by a $\\sigma$-derivation $d_\\sigma$ that satisfies a twisted Leibniz rule $d_\\sigma(f g)=d_\\sigma(f)g+\\sigma(f) d_\\sigma(g) $, where $\\sigma$ is an algebra endomorphism of a commutative associative algebra. An example of a $\\sigma$-derivation is the Jackson derivative on polynomials in one variable. A general construction of quasi-Lie algebras and the introduction of Hom-Lie algebras were given in \\cite{HLS}. The corresponding associative algebras, called Hom-associative algebras, were introduced in \\cite{MS}, where it is shown that they are Hom-Lie admissible, while the enveloping algebra of a Hom-Lie algebra was constructed in \\cite{Yau08}. Moreover, Hom-bialgebras and Hom-Hopf algebras were studied in \\cite{MS2009,MS2010a,Yau4}. Further results could be found in \\cite{AM,AEM,BEM,E,LS,MS2,MS2010,Sh,Yau09,Yau10}.\n\n\nThe purpose of this article is two-fold. First, we construct explicitly the universal enveloping algebra of a multiplicative Hom-Lie algebra and show that it is a Hom-Hopf algebra. Then, we intend to mimic the construction of a Lie group integrating a Lie algebra $\\gg$ obtained by choosing, as a candidate for integrating the Lie algebra $\\gg$, group-like elements of the universal enveloping algebra $ {\\mathcal U}\\gg$.\n\nThis task, in particular the second step, is not trivial, and forced us to reconsider the way antipodes are generally defined on bialgebras, as well as the definition of invertible elements on Hom-groups. Still, we are able to present a Hom-group that we claim to be the integration of a Hom-Lie algebra, but it is more involved than simply group-like elements in the universal enveloping algebra of a Hom-Lie algebra.\nBefore describing our construction step by step, let us discuss what inverse means in the context of Hom-algebras.\n\n\n\\subsection*{Invertibility and inverse on Hom-algebras or Hom-groups}\n\nThere is one natural manner, which was already considered by \\cite{Yau08}, to construct the universal enveloping algebra $ {\\mathcal U}\\gg$ of a multiplicative Hom-Lie algebra $\\gg$. This object is, as expected, a Hom-bialgebra. But, as we shall see while proving Theorem \\ref{thm:HopfAlgOnTrees}, it turns out \\emph{not} to be a Hom-Hopf algebra in the sense of \\cite{MS2010a}, because the antipode is not an inverse of the identity map for the convolution product.\n\nThis failure is, however, productive, in the sense that it paves the way for what seems to be a definition of a Hom-Hopf algebra suitable in this context. Let us explain the situation. In the Hom-bialgebra $ ({\\mathcal U}\\gg,\\vee, \\alpha,\\Delta, \\eta, \\epsilon)$ the usual antipode $S$ (for Hopf algebra structure) does not satisfy the axiom:\n$$ S \\star id = id \\star S = \\eta \\circ \\epsilon $$\nwith $\\star $ being the convolution product, defined by $S \\star T = \\vee \\circ (S \\otimes T) \\circ \\Delta $\nfor two arbitrary linear endomorphisms $S,T$ of ${\\mathcal U}\\gg$, as in the classical case. The antipode $S$ only satisfies a weakened condition: for any $ x \\in {\\mathcal U}\\gg$, there exists an integer $ k \\in {\\mathbb N}$ such that:\n \\begin{equation}\\label{eq:weaker} \\alpha^k \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta \\, (x) = \\alpha^k \\circ \\vee \\circ (id \\otimes S) \\circ \\Delta \\, (x) = \\eta \\circ \\epsilon\\, (x) .\n\t\t\t\\end{equation}\nThe smallest such integer $k$ is called the invertibility index of $x$.\nWe define Hom-Hopf algebras as bialgebras satisfying this weakened condition.\n\nThis definition is indeed not surprising. Given a Hom-associative algebra $ ( {\\A} , \\vee , \\alpha )$ that admits a unit ${\\mathds{1}} $, it is tempting to define invertible elements as being elements $x$ in $ {\\A}$ such that there exists $ y \\in {\\A}$ with $ x \\vee y=y \\vee x= {\\mathds{1}}$.\nAlternatively, it may be tempting to define invertible elements to be those elements $x$ in $ {\\A}$ such that there exists $ y \\in {\\A}$ with $ \\alpha(x \\vee y)=\\alpha(y \\vee x)= {\\mathds{1}}$\nas in \\cite{F}. However, there is an issue with both definitions: invertible elements in any of these two senses are in general not stable under $\\vee $.\n\nIn order to get a notion of invertible elements that would allow those to be invariant under $\\vee $, we say that an element $x \\in {\\A}$ is \\textbf{hom-invertible} if and only if there exists $y \\in {\\A}$ (not necessarily unique) called a \\textbf{hom-inverse} and an integer $k \\in {\\mathbb N}$ such that\n$$ \\alpha^k ( x \\vee y ) = \\alpha^k ( y \\vee x ) = {\\mathds{1}} .$$\nThis definition is consistent with Equation (\\ref{eq:weaker}). The antipode $S$ that we have constructed on the Hom-bialgebra $ ({\\mathcal U}\\gg,\\vee, \\Delta, \\eta, \\epsilon)$ becomes now a kind of hom-inverse of the identity for the convolution product, making $ ({\\mathcal U}\\gg,\\vee, \\Delta, \\eta, \\epsilon, S)$ a Hom-Hopf algebra.\n\nThis issue being solved, we intend to find a Hom-group integrating a Hom-Lie algebra. Having modified the definition of a hom-inverse, we have to modify the definition of Hom-Lie group accordingly.\n\n\\begin{defn}\\label{def:ourHomGroup}\nA \\textbf{Hom-group} is a set $(G,\\vee,\\alpha,{\\mathds{1}})$ equipped with a Hom-associative product with unit ${\\mathds{1}} $ and an anti-morphism $g \\to g^{-1}$ such that, for any $g \\in G$, there exists an integer $k \\in {\\mathbb N}$ satisfying\n$$ \\alpha^k ( g\\vee g^{-1}) = \\alpha^k ( g^{-1} \\vee g ) = {\\mathds{1}} .$$\nThe smallest such integer $k$ is called the invertibility index of $g$.\n\\end{defn}\n\n\n\\subsection*{From Hom-Lie algebras to Hom-groups}\n\nGroup-like elements in a Hom-Hopf algebra $ {\\A}$ (equipped with an antipode satisfying the weakened assumption (\\ref{eq:weaker})) form a Hom-group (group-like elements being defined as formal series $ g(\\nu )\\in {\\A}[[\\nu]]$\nwith some assumptions). It is therefore tempting to define the object integrating the Hom-Lie algebra $ \\gg$ as being the set of group-like elements in the Hom-Hopf algebra $ {\\mathcal U}\\gg[[\\nu]]$. However, this definition is irrelevant: there is in general very little group-like elements in $ {\\mathcal U}\\gg[[\\nu]]$, except for the unit $ {\\mathds{1}}$ itself.\n\nIt is possible, fortunately, to go around this difficulty by defining, for all $p \\in {\\mathbb N}$, {$p$-order formal group-like elements} as being elements in $ {\\mathcal U}\\gg[[\\nu]]$ satisfying\n$ g(0)={\\mathds{1}} $ and:\n $$ \\Delta g(\\nu) = g(\\nu) \\otimes g(\\nu) \\hspace{1cm} \\left[\\nu^{p+1}\\right]$$\n(where $\\left[\\nu^{p+1}\\right]$ means \"modulo $ \\nu^{p+1} $\").\nIt is routine to check that $p$-order formal group-like element do form a Hom-group, with inverse given by the antipode.\nThen we consider sequences $ (g_p(\\nu))_{p \\geq 0}$, with $g_p(\\nu)$ a $p$-order formal group-like element,\nsuch that the quotient of $ g_{p+1} (\\nu)$ modulo $ \\nu^{p+1}$ is $\\alpha ( g_p(\\nu) ) $ for all $p \\in {\\mathbb N}$.\n We call these sequences {formal group-like sequences} when their invertibility index is bounded. We show that formal group-like sequences do form a Hom-group, with inverse again induced by the antipode. Moreover an exponential map valued in formal group-like sequence can be constructed making this Hom-group a reasonable candidate for being considered as a Hom-group integrating the Hom-Lie algebra $ \\gg$, as several theorems will show at the end of this article.\n\n\n\\section{Hom-Lie algebras, Hom-associative algebras and Hom-bialgebras}\n\nGiven $\\gg$ a vector space and a bilinear map $ \\brr{\\, , \\, }: \\gg \\otimes \\gg \\to \\gg$, by \\textbf{endomorphism of\n$(\\gg,\\brr{\\, , \\, } )$}, we mean a linear map $\\alpha: \\gg \\to \\gg$ such that\n$$\n\\alpha (\\brr{x,y}) = \\brr{\\alpha (x), \\alpha (y)}\n$$\nfor all $x,y \\in \\gg$. We now define Hom-Lie algebras, sometimes called multiplicative Hom-Lie algebras.\n\n\\begin{defn}\\label{def:hom:Lie:algebra} \\cite{HLS}\nA \\textbf{Hom-Lie algebra} is a triple $(\\gg, \\brr{\\, , \\, }, \\alpha)$ with $\\gg$ a vector space equipped with a\nskew-symmetric bilinear map $ \\brr{\\, , \\, }:\\gg \\otimes \\gg \\to \\gg$ and an endomorphism\n$\\alpha$ of $(\\gg,\\brr{\\, , \\, })$ such that:\n\\begin{equation}\n\\brr{\\alpha (x),\\brr{y,z}}+\\brr{\\alpha (y),\\brr{z,x}}+\\brr{\\alpha\n(z),\\brr{x,y}}=0, \\quad \\forall x,y,z\\in\\gg \\quad \\hbox{(Hom-Jacobi identity)}. \\label{eq:hom:Jacobi:algebra}\n\\end{equation}\nA \\textbf{morphism} between Hom-Lie algebras $(\\gg,\\brr{\\, , \\, }_\\gg,\\alpha)$ and $(\\hh,\\brr{\\, , \\, }_\\hh,\\beta)$ is a linear map $\\psi:\\gg \\to \\hh$\nsuch that $\\displaystyle \\psi(\\brr{x,y}_\\gg)=\\brr{\\psi( x),\\psi (y)}_\\hh$ and $\\psi (\\alpha(x))=\\beta(\\psi(x))$\nfor all $x,y \\in \\gg$.\nWhen $\\hh$ is a vector subspace of $\\gg$ and $\\psi$ is the inclusion map, one speaks of \\textbf{Hom-Lie subalgebra}.\n\\end{defn}\n\n\\begin{rem}\\label{Rmk:IdealAndInverse}\nLet $(\\gg, \\brr{\\, , \\, }, \\alpha)$ be a Hom-Lie algebra. The subspace ${\\mathfrak k} \\subset \\gg$ of elements $x \\in \\gg$ such that there exists an integer $k$ with $\\alpha^k (x)=0$ is a Hom-Lie subalgebra. It is even a Hom-Lie ideal, i.e. the quotient $\\gg\/{\\mathfrak k} $ inherits a structure of Hom-Lie algebra. By construction, its induced morphism $\\underline{\\alpha}: \\gg\/{\\mathfrak k} \\to \\gg\/{\\mathfrak k}$ is invertible. The induced bracket $\\underline{\\alpha}^{-1} \\circ \\underline{\\brr{\\, , \\, }}$ is therefore a Lie algebra bracket, see \\cite{G}. Also, the natural projection $\\gg\/{\\mathfrak k} $ is a morphism of Hom-Lie algebra.\n\\end{rem}\n\n\n\n\\begin{defn} \\label{hom-associative}\\cite{MS}\nA \\textbf{Hom-associative algebra} is a triple $(\\A, \\mu, \\alpha)$ consisting of a vector space $\\A$, a bilinear map $\\mu: \\A \\otimes \\A \\to \\A$ and an endomorphism $\\alpha$ of $( \\A, \\mu) $ satisfying\n$$\n\\mu(\\alpha(x), \\mu(y,z))= \\mu (\\mu(x,y), \\alpha (z)), \\quad \\forall x,y,z \\in \\A \\hbox{ (Hom-associativity)}.\n$$\nA Hom-associative algebra $(\\A, \\mu, \\alpha)$ is called \\textbf{unital} if there exists a linear map $\\eta: \\mathbb K \\to \\A$ such that\n$$\n\\mu \\circ (id_{\\A} \\otimes \\eta) = \\mu \\circ (\\eta \\otimes id_{\\A} )= \\alpha \\hbox{ and } \\alpha \\circ \\eta = \\eta.\n$$\nWe denote a unital Hom-associative algebra by a quadruple $(\\A, \\mu, \\alpha, \\eta)$. The unit element (or unit, for simplicity) is $\\mathds{1}= \\eta(1_{\\mathbb K})$.\n\\end{defn}\n\nNotice that a Hom-associative algebra $(\\A, \\mu, \\alpha)$ is unital, with unit $\\mathds{1} \\in \\A$,\nwhen $ \\mu(x, \\mathds{1})= \\mu(\\mathds{1},x) =\\alpha(x) $ and $ \\alpha(\\mathds{1})=\\mathds{1}$.\n\nMorphisms between Hom-associative algebras are defined in the similar way as Hom-Lie algebras. For unital Hom-associative algebras, the image of the unit is a unit.\n\\begin{ex}\\cite{Yau09,MS}\n\\label{ex:composition}\nGiven a vector space $\\gg$ equipped with a bilinear map $ \\brr{\\, , \\, }:\\gg \\otimes \\gg\n\\to \\gg$ and an endomorphism $\\alpha:\\gg\\to \\gg$ of $(\\gg, \\brr{\\, , \\, })$. Define\n$ \\brr{\\, , \\, }_{\\alpha}:\\gg \\otimes \\gg\n\\to \\gg$ by\n$$ \\brr{x,y}_\\alpha=\\alpha (\\brr{x,y}), \\quad \\hbox{ $\\forall x,y \\in \\gg$.} $$\n Then $(\\gg, \\brr{\\, , \\, }_{\\alpha}, \\alpha)$ is a Hom-Lie algebra (resp. a Hom-associative algebra, resp. unital Hom-associative algebra) if and only if the restriction of $\\brr{\\, , \\, }$\nto the image of $\\alpha^2 $ is a Lie bracket (resp. an associative product, resp. a unital associative product).\nIn particular, Hom-Lie structures are naturally associated to Lie algebras equipped with a Lie algebra endomorphism \\cite{Yau09}.\nSuch Hom-Lie structures are said to be \\textbf{obtained by composition} or \\textbf{Twisting principle}.\n\\end{ex}\n\n\n\\begin{ex} \\label{ex:composition2}\nAs one can expect, the commutator of a Hom-associative algebra is a Hom-Lie algebra\n\\cite{MS}. More precisely, for every Hom-associative algebra $(\\A, \\mu, \\alpha)$ (see Definition \\ref{hom-associative} above), the triple $(\\A, \\brr{\\, , \\, }, \\alpha)$ is a Hom-Lie algebra, where\n$$\n\\brr{x,y}:= \\mu(x,y)- \\mu (y,x)\n$$\nfor all $x,y \\in \\A$.\n\\end{ex}\n\n\\begin{defn}\\label{def:our_inverse}\nAn element $x$ in a unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $ is said to be \\textbf{hom-invertible} if there exists an element $x^{-1}$ and a non-negative integer $k \\in {\\mathbb N}$, such that\n\\begin{equation}\\label{eq:def_inverse}\n\\alpha^k \\circ \\mu(x, x^{-1})= \\alpha^k \\circ \\mu(x^{-1}, x) = \\mathds{1}.\n\\end{equation}\n The element $x^{-1}$ is called a \\textbf{hom-inverse} and the smallest $k$ is the \\textbf{invertibility index} of $x$.\n\\end{defn}\nWhen it exists, the hom-inverse may not be unique, which prevents hom-invertible elements to be a Hom-group in the sense of Definition \\ref{def:ourHomGroup}. However, the following can be shown.\n\n\\begin{prop}\\label{prop:stableByProduct}\nFor every unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $, the unit $\\mathds{1}$ is hom-invertible,\nthe product of any two hom-invertible elements is hom-invertible and every inverse of a hom-invertible element is hom-invertible.\n \\end{prop}\n\\begin{proof}\n The only non-trivial point is that $\\mu(x , x')$ is a hom-invertible element if both $x$ and $x'$ are, i.e.\n if there exists $y,y' \\in \\A$, $k,k' \\in {\\mathbb N}$ such that\n $$\\alpha^k \\circ \\mu(x, y)= \\alpha^k \\circ \\mu(y,x) = \\alpha^{k'} \\circ \\mu(x', y')= \\alpha^{k'} \\circ \\mu(y', x') = \\mathds{1}.$$\n Hom-associativity implies\n $$ \\alpha \\circ \\mu(\\mu(x,x') , \\mu(y',y)) = \\mu( \\alpha^2(x) , \\mu(\\mu(x',y'), \\alpha(y) ) .\n $$\n\n So that\n \\begin{eqnarray*} \\alpha^{k+k'+1} \\circ \\mu(\\mu(x,x') , \\mu(y',y)) & = & \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\mu(\\alpha^{k'} \\mu(x',y'), \\alpha^{k'+1}(y) )\n \\\\ & =& \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\mu(\\mathds{1}, \\alpha^{k'+1}(y) ) \\\\\n & =& \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\alpha^{k'+2}(y) ) \\\\ &=& \\alpha^{k+k'+2} ( \\mu( x , y ) ) \\\\\n &=& \\alpha^{k'+2}(\\mathds{1}) = \\mathds{1}.\n \\end{eqnarray*}\n This completes the proof.\n\\end{proof}\n\n\\begin{rem}\nFor every unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $, the subspace ${\\mathfrak k}$ of all elements $x \\in \\A$ such that $ \\alpha^k (x)=0$ for some integer $k \\in {\\mathbb N}$ is an ideal, i.e. the quotient map\n$\\A\/{\\mathfrak k}$ is a unital Hom-associative algebra for which the induced map $\\underline{\\alpha}$ is invertible\n for the induced product $ \\underline{\\mu}$.\nIn particular $ \\A\/{\\mathfrak k}$ equipped with the product $\\underline{\\alpha}^{-1} \\circ \\underline{\\mu} $ is an algebra.\n\nAn element $x\\in \\A$ is invertible in $(\\A, \\mu, \\alpha,\\mathds{1}) $ if and only if its image in $\\A\/{\\mathfrak k}$ is invertible in the usual sense, which gives an alternative proof of Proposition \\ref{prop:stableByProduct}.\n\\end{rem}\n\nWe now recall the notion of Hom-coalgebras.\n\n\n\n\n\n\\begin{defn} \\cite{MS}\nA \\textbf{Hom-coalgebra} is a triple $(A, \\Delta, \\beta)$ where $A$ is a vector space and $\\Delta: A \\to A\\otimes A$, $\\beta: A \\to A$ are linear maps.\n\nA \\textbf{Hom-coassociative coalgebra} is a Hom-coalgebra $(A, \\Delta, \\beta)$ satisfying\n$$\n(\\beta \\otimes \\Delta) \\circ \\Delta=(\\Delta \\otimes \\beta) \\circ \\Delta.\n$$\nA Hom-coassociative coalgebra is said to be \\textbf{co-unital} if there exists a linear map $\\epsilon: A \\to \\mathbb K$ satisfying\n$$\n(id \\otimes \\epsilon) \\circ \\Delta= \\beta \\hbox{, } (\\epsilon \\otimes id) \\circ \\Delta = \\beta\n \\hbox{ and } \\epsilon \\circ \\beta = \\epsilon.\n$$\nWe refer to a counital Hom-coassociative coalgebra with a quadruple $(A,\\Delta, \\beta, \\epsilon)$.\n\\end{defn}\n\nLet $(A, \\Delta, \\beta)$ and $(A', \\Delta', \\beta')$ be two Hom-coalgebras (resp. Hom-coassociative algebras). A linear map $f: A \\to A'$ is a morphism of Hom-coalgebras (resp. Hom-coassociative coalgebras) if\n$$\n(f \\otimes f) \\circ \\Delta = \\Delta' \\circ f \\qquad f \\circ \\beta = \\beta' \\circ f.\n$$\nIt is said to be a weak Hom-coalgebras morphism if it holds only the first condition. If furthermore the Hom-coassociative coalgebras admit counits $\\epsilon$ and $\\epsilon'$, we have moreover $\\epsilon = \\epsilon' \\circ f$.\n\nThe category of coassociative Hom-coalgebras is closed under weak Hom-coalgebra morphisms.\n\n\\begin{ex}\n\\cite{MS2009}\nThe dual of a Hom-algebra $(A,\\mu,\\alpha)$ is not always a Hom-coalgebra, because the coproduct does not land in the good space $\\mu^*: A^* \\to (A \\otimes A)^* \\supsetneq A^* \\otimes A^*$.\nNevertheless, it is the case if the Hom-algebra is finite-dimensional, since $(A \\otimes A)^*= A^* \\otimes A^*$.\nThe converse always holds true. Let $(A, \\Delta, \\beta)$ be a Hom-coassociative coalgebra. Then its dual vector space is provided with a structure of Hom-associative algebra $(A^*, \\Delta^*, \\beta^*)$ where $\\Delta^*$, $\\beta^*$ are the transpose maps.\nMoreover, the Hom-associative algebra is unital whenever $A$ is counital.\n\n\n\\cite{MS2010a} Let $(A,\\Delta, \\beta, \\epsilon)$ be a counital Hom-coassociative coalgebra and $\\alpha: A \\to A$ be a weak Hom-coalgebra morphism. Then $(A,\\Delta_{\\alpha}= \\Delta \\circ \\alpha, \\beta \\circ \\alpha, \\epsilon)$ is a counital Hom-coassociative coalgebra.\nIn particular, let $(A,\\Delta,\\epsilon)$ be a coalgebra and $\\beta: A \\to A$ be a coalgebra morphism. Then $(A,\\Delta_{\\beta}, \\beta, \\epsilon)$ is a counital Hom-coassociative coalgebra\n\\end{ex}\n\n\n\n\\begin{defn} \\cite{MS2009}\nAn $(\\alpha,\\beta)$-\\textbf{Hom-bialgebra} (simply called Hom-bialgebra when there is no ambiguity) is a heptuple $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ where\n\\begin{enumerate}\n \\item[(i)] $(A,\\mu,\\alpha, \\eta)$ is a Hom-associative algebra with unit $\\eta$ and unit element $\\mathds{1}$,\n \\item[(ii)] $(A,\\Delta, \\beta, \\epsilon)$ is a Hom-coassociative coalgebra with a counit $\\epsilon$,\n \\item[(iii)] the linear maps $\\Delta$ and $\\epsilon$ are compatible with the multiplication $\\mu$ and the unit $\\eta$, that is for $x,y \\in A$\n\\begin{enumerate}\n \\item $\\displaystyle{\\Delta(\\mu(x \\otimes y))= \\Delta(x) \\cdot \\Delta(y)= \\sum_{(x)(y)} \\mu(x_1 \\otimes y_1) \\otimes \\mu(x_2 \\otimes y_2)}$, where $\\cdot$ denotes the multiplication on the tensor algebra $A \\otimes A$,\n \\item $\\Delta(\\mathds{1})= \\mathds{1} \\otimes \\mathds{1}$,\n \\item $\\epsilon(\\mathds{1})= 1$,\n \\item $\\epsilon(\\mu(x \\otimes y))= \\epsilon(x) \\epsilon(y)$,\n \\item $\\epsilon \\circ \\alpha(x)= \\epsilon(x)$.\n\\end{enumerate}\n\\end{enumerate}\nIf $\\alpha=\\beta$ the $(\\alpha,\\alpha)$-Hom-bialgebra is denoted by the hexuple $(A, \\mu, \\eta, \\Delta, \\epsilon, \\alpha)$.\n\\end{defn}\n\nA Hom-bialgebra morphism is a linear endomorphism which is simultaneously a Hom-algebra and Hom-coalgebra morphism.\n\n\\begin{ex}\\label{TwistingPrincipleBialgebra}\nLet $(A,\\mu, \\eta,\\Delta,\\epsilon, \\alpha)$ be a Hom-bialgebra and $\\beta:A \\to A$ be a Hom-bialgebra morphism. Then $(A, \\mu_{\\beta} = \\beta \\circ \\mu, \\eta,\\Delta_{\\beta} = \\Delta \\circ \\beta, \\epsilon, \\beta \\circ \\alpha)$ is a Hom-bialgebra.\nIn particular, if $(A,\\mu, \\eta,\\Delta, \\epsilon)$ is a bialgebra and $\\beta: A \\to A$ is a bialgebra morphism then $(A, \\mu_{\\beta}, \\eta,\\Delta_{\\beta}, \\epsilon, \\beta )$ is a Hom-bialgebra.\nThis construction method of Hom-bialgebra, starting with a given Hom-bialgebra or a bialgebra and a morphism, is called composition method or twisting principle \\cite{MS2010a}. We can also define an $(\\alpha, \\beta)$ twist. If $(A,\\mu, \\eta,\\Delta, \\epsilon)$ is a bialgebra and $\\alpha, \\beta: A \\to A$ are bialgebra morphisms which commute, that is $\\alpha \\circ \\beta = \\beta \\circ \\alpha$,\nthen $(A, \\mu_{\\alpha}=\\alpha \\circ \\mu, \\alpha,\\eta,\\Delta_{\\beta}=\\Delta \\circ \\beta, \\beta,\\epsilon )$ is a Hom-bialgebra. In particular we can consider one of the morphisms equal to identity.\n\\end{ex}\n\n\n\\section{Hom-Hopf algebras}\n\nThe following theorem holds and the proof goes through a direct verification of the axioms.\n\n\\begin{thm} \\label{conv-product}\n\\cite{MS2009,MS2010a}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ be an $(\\alpha, \\beta)$-Hom-bialgebra. Then $({\\rm Hom}(A,A),\\star,\\gamma)$ is a unital Hom-associative algebra, with $\\star$ being the multiplication given by the convolution product defined by\n$$\nf\\star g = \\mu \\circ (f \\otimes g) \\circ \\Delta\n$$\nand $ \\gamma$ being the homomorphism of ${\\rm Hom}(A,A)$ defined by $\\gamma(f)= \\alpha \\circ f \\circ \\beta$. The unit is $\\eta \\circ \\epsilon$.\n\\end{thm}\n\n\n\n\nWe now define Hom-Hopf algebra in a manner that differs from \\cite{MS2009,MS2010a}. In those works, an antimorphism $S$ of $A$ is said to be an antipode if it is an inverse (in the usual sense) of the identity\nover $A$ for the Hom-associative algebra ${\\rm Hom} (A,A)$ with the multiplication given by the convolution product, i.e.\n $ S \\star id =id \\star S = \\eta \\circ \\epsilon $.\nThis definition matches examples given by twisting principle out of a Hopf algebra but does not match our examples.\nFor this reason, and also in view of the Definition \\ref{def:our_inverse} of an invertible element in the context of Hom-algebras, we prefer to define an antipode as being an anti-morphism $S$ of $A$ which is a relative hom-inverse of the identity, defined as follows. We say that $S \\in {\\rm Hom} (A,A)$ is a \\textbf{relative hom-inverse} of $T \\in {\\rm Hom} (A,A)$ if and only if for every $x \\in A$ there exists an integer $k$ (depending on $x$) such that:\n$$ \\alpha^{k} (S \\star T ) \\, (x) = \\alpha^k (T \\star S ) \\, (x) = \\eta \\circ \\epsilon \\, (x).$$\n\n\\begin{rem}\nNotice that $S$ is not a hom-inverse of $T$ in the sense of Definition \\ref{def:our_inverse}.\nHowever, if there exists an integer\n$\\bar{k}$ such that $ \\alpha^{\\bar{k}} (S \\star T ) \\, (x) = \\alpha^{\\bar{k}} (T \\star S ) \\, (x) = \\eta \\circ \\epsilon \\, (x)$ for all $x\\in A$, then $S$ is a hom-inverse of $T$ and the smallest $\\overline{k}$ is the invertibility index of $S$.\\end{rem}\n\n\nThis amounts to the following definition:\n\n\\begin{defn} \\label{def:hom-Hopf-algebra_in_our_sense}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ be an $(\\alpha, \\beta)$-Hom-bialgebra.\nAn anti-homomorphism $S$ of $A$ is said to be an \\textbf{antipode} if\n\\begin{enumerate}\n\\item[a)] $ S \\circ \\alpha = \\alpha \\circ S$,\n\\item[b)] $ S \\circ \\eta= \\eta $ and $ \\epsilon \\circ S = \\epsilon$,\n\\item[c)] $S$ is a relative Hom-inverse of the identity map $id :A \\to A$ for the convolution product given as in Theorem \\ref{conv-product}.\n\\end{enumerate}\n An $(\\alpha, \\beta)$-\\textbf{Hom-Hopf algebra} is an $(\\alpha, \\beta)$-Hom-bialgebra admitting an antipode.\n\\end{defn}\n\nRecall that condition (c) means equivalently that, for every $x \\in A $, there exists $ k \\in {\\mathbb N}$ such that:\n \\begin{equation}\\label{eq:antipode_condition}\\alpha^k \\circ( S \\star id) (x)= \\alpha^k \\circ (id \\star S) (x) = \\eta \\circ \\epsilon (x).\n \\end{equation}\n\n\nNotice that we do not need to assume that $ S $ and $ \\beta $ commute (in most examples,\n$\\beta$ is either the identity or coincides with $\\alpha$).\n\n\\begin{ex} \\label{Hom-Hopf-Twist}\nLet $(A,\\mu, \\eta,\\Delta, \\epsilon, S)$ be a Hopf algebra, and $\\alpha, \\beta: A \\to A$ be commuting bialgebra morphisms satisfying $S\\circ \\alpha=\\alpha \\circ S$. Then\n$$\n(A, \\mu_{\\alpha}=\\alpha\\circ \\mu,\\alpha, \\eta,\\Delta_{\\beta}=\\Delta\\circ\\beta, \\beta,\\epsilon ,S)\n$$\nis an $(\\alpha, \\beta)$-Hom-Hopf algebra, called the \\textbf{$(\\alpha,\\beta)$-twist} of the Hopf algebra $A$.\nMore generally, the same idea turns a $(\\alpha', \\beta')$-Hom-Hopf algebra in a $(\\alpha \\circ \\alpha', \\beta \\circ \\beta')$-Hom-Hopf algebra.\n\nIndeed, according to Example \\ref{TwistingPrincipleBialgebra}, $\n(A, \\mu_{\\alpha}=\\alpha\\circ \\mu,\\alpha, \\eta,\\Delta_{\\beta}=\\Delta\\circ\\beta, \\beta,\\epsilon )\n$ is a Hom-bialgebra. It remains to show that $S$ is still an antipode for the Hom-bialgebra. We have\n$$S(\\mu_\\alpha(x,y))=S(\\alpha(\\mu(x,y)))=\\alpha(S(\\mu(x,y)))=\\alpha(\\mu(S(y),S(x)))=\\mu_\\alpha(S(y),S(x)),\n$$\nand\n$$\\mu_\\alpha\\circ(S\\otimes \\id)\\circ\\Delta_\\beta=\\alpha\\circ\\mu\\circ(S\\otimes \\id)\\circ\\Delta\\circ\\beta=\n\\mu\\circ(S\\otimes \\id)\\circ\\Delta\\circ\\alpha\\circ\\beta=\\eta\\circ\\epsilon\\circ\\alpha\\circ\\beta=\\eta\\circ\\epsilon,\n$$\nwhich complete the proof. The proof for the general case is similar.\n\\end{ex}\n\n\n\n\n\\begin{rem}\n\\label{rmk:SeveralPoints}\nFor every $(\\alpha,\\beta)$-Hom-Hopf algebra, the following properties hold:\n\\begin{enumerate}\n \\item Using counitality, we have (in Sweedler's notation):\n $ \\beta(x) = \\sum x_1 \\epsilon (x_2) = \\sum \\epsilon (x_1) \\, x_2$.\n \\item Let $x$ be a primitive element (which means that $\\Delta(x)= \\mathds{1}\\otimes x + x \\otimes \\mathds{1}$), then $\\epsilon(x)= 0$.\n \\item If $x$ and $y$ are two primitive elements in $A$, then we have $\\epsilon(x)=0$ and the commutator $[x,y]= \\mu(x \\otimes y) - \\mu (y \\otimes x)$ is also a primitive element.\n \\item The set of all primitive elements of $A$, denoted by ${\\rm Prim}(A)$, admits a natural structure of Hom-Lie algebra, with bracket given by the commutator $ [x,y]:= \\mu(x \\otimes y) - \\mu (y \\otimes x)$, see \\cite{MS2009,MS2010a}.\n \\item If $x,y,z$ are primitive elements in $A$, then the Hom-associator $\\mu(\\alpha(x),\\mu(y,z))-\\mu(\\mu(x,y),\\alpha(z))$ is a primitive element.\n\t \\item Using counitality and unitality, we have $ S \\star (\\eta \\circ \\epsilon) = \\alpha \\circ S \\circ \\beta =\\gamma(S) $,\n\tand more generally $ (\\alpha^p \\circ S \\circ \\beta^q )\\star (\\eta \\circ \\epsilon) = \\alpha^{p+1} \\circ S \\circ \\beta^{q+1} $.\n\t\\item For all linear endomorphism $ S,T $ of $ A$, we have $\\alpha ( S \\star T) = (\\alpha \\circ S) \\star (\\alpha \\circ T) $.\n\t\\item The antipode condition (\\ref{eq:antipode_condition}) can be stated as $ (\\alpha^k \\circ S) \\star \\alpha^k = \\alpha^k \\star (\\alpha^k \\circ S) = \\eta \\circ \\epsilon$.\n\\end{enumerate}\n\\end{rem}\n\n\n\n\n\\begin{prop}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon) $ be an $(\\alpha,\\beta)$-Hom-bialgebra. Assume that $S $ and $S'$ are two antipodes. Let $x\\in A$ and $k,k' \\in {\\mathbb N}$\n such that:\n $$ \\alpha^{k} ( S \\star \\id )(x) = \\alpha^{k} ( \\id \\star S ) (x)= \\eta \\circ \\epsilon(x) \\hbox{ and }\n\t\\alpha^{k'} ( S' \\star \\id )(x) = \\alpha^{k'} ( \\id \\star S' )(x) = \\eta \\circ \\epsilon(x).$$\n\tThen, the following relation holds\n $$ \\alpha^{K+2} \\circ S \\circ \\beta^2(x) = \\alpha^{K+2} \\circ S' \\circ \\beta^2 (x) $$\nwith $K ={\\rm max}(k,k')$.\n\\end{prop}\n\\begin{proof}We assume that $k'\\leq k$.\nRecall that for any $f$, $f\\star \\eta\\circ \\epsilon =\\alpha \\circ f\\circ \\beta$. For simplicity, we omit the composition circle.\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&=\\alpha(\\alpha^{k+1}S'\\beta)\\beta=(\\alpha^{k+1}S'\\beta)\\star (\\eta \\epsilon)=(\\alpha^{k+1}S'\\beta)\\star(\\alpha^k(\\id \\star S))=(\\alpha^{k+1}S'\\beta)\\star(\\alpha^k \\star \\alpha^k S).\n\\end{align*}\nBy Hom-associativity we have\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&=(\\alpha^{k}S'\\star\\alpha^k)\\star(\\alpha^{k+1} S\\beta)\n=(\\alpha^{k-k'}(\\alpha^{k'}(S'\\star\\id))\\star(\\alpha^{k+1} S\\beta)\n=(\\alpha^{k-k'}\\eta\\epsilon))\\star(\\alpha^{k+1} S\\beta).\n\\end{align*}\nSince $\\alpha\\eta=\\eta$ and $\\epsilon\\beta=\\epsilon$, we have\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&\n=(\\eta\\epsilon)\\star(\\alpha^{k+1} S\\beta)=(\\alpha^{k+1}\\eta\\epsilon\\beta)\\star(\\alpha^{k+1} S\\beta)=\\alpha^{k+1}(\\eta\\epsilon\\star S)\\beta=\\alpha^{k+2}S\\beta^2.\n\\end{align*}\n\\end{proof}\n\\begin{rem}\nThis proposition means that the antipode is in some sense unique.\nIndeed, when $\\alpha$ and $\\beta$ are invertible, the antipode is unique when it exists.\n\\end{rem}\n\n\n\\section{Elements of group-like type in an $(\\alpha,\\beta)$-Hom-Hopf algebra}\n\\label{sec:grouplike}\n\nFor $( {A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta, \\epsilon, S) $ an $(\\alpha,\\beta)$-Hopf-algebra, all the structural maps\nextend by ${\\mathbb K}[[\\nu]] $-linearity to yield an $(\\alpha,\\beta)$-Hom-bialgebra structure on the space ${A}[[\\nu]]$ of formal series with coefficients in $A$. However, it may not be an $(\\alpha,\\beta)$-Hom-Hopf algebra because\n$S$ may not be a relative-inverse of the identity map.\nHowever, for formal series $g(\\nu)=\\sum_{i \\geq 0} g_i \\nu^i$ such that the invertibility indexes of the elements $(g_i)_{i \\in {\\mathbb N}}$ are bounded, there exits $k\\in {\\mathbb N}$ such that\n$$ \\alpha^k\\circ(S\\star \\id )(g_i)= \\alpha^k\\circ(\\id \\star S )(g_i)=\\eta \\circ \\epsilon (g_i).$$\nSumming up these relations, we obtain:\n\n\n\\begin{prop}\nThe space $A_{b}[[\\nu]]$ of formal series $g(\\nu)=\\sum_{i \\geq 0} g_i \\nu^i$, such that the invertibility indexes of the elements $(g_i)_{i \\in {\\mathbb N}}$ are bounded,\nis an $(\\alpha,\\beta)$-Hom-Hopf algebra.\n\\end{prop}\nIn particular, polynomial elements are in $ A_{b}[[\\nu]] $.\n\n\n\\begin{defn}\n\\label{def:groupLikeAndTheLikes}\nLet $( {\\A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta , \\epsilon, S) $ be an $(\\alpha,\\beta )$-Hopf-algebra.\nAn element $g\\in \\A$ is a {\\textbf{ group-like element}} if\n$ \\Delta (g) = g \\otimes g \\hbox{ and } \\epsilon(g)=1$.\nLet $\\nu $ be a formal parameter.\n\\begin{enumerate}\n \\item[(i)] We call a {\\textbf{formal group-like element}} a formal series $g(\\nu) \\in {\\A}_b[[\\nu]] $ such that:\n\\begin{equation}\\label{eq:group-like} \\Delta (g(\\nu)) = g(\\nu) \\otimes g(\\nu) \\hbox{ and } \\epsilon(g(\\nu))=1 .\\end{equation}\n \\item[(ii)]\nElements in ${A}[\\nu] $ (i.e. polynomials in $\\nu$) are called {\\textbf{$p$-order group-like elements}}\nwhen:\n\\begin{equation}\\label{eq:group-like-order-n} \\Delta (g(\\nu)) = g(\\nu) \\otimes g(\\nu) \\hbox{ modulo } \\nu^{p+1} \\hbox{ and } \\epsilon(g(\\nu))=1.\\end{equation}\n \\item [(iii)]\nA {\\textbf{formal group-like sequence}} is a sequence $ (g_p (\\nu))_{p \\in {\\mathbb N}} $ of elements in ${\\A}[\\nu] $ such that\n \\begin{enumerate}\n \\item[a)] for all $p \\geq 1$, $g_p(\\nu) $ is a $p$-order formal group-like element,\n \\item[b)] $g_{p+1} (\\nu)= \\alpha (g_p (\\nu))$ modulo $\\nu^{p+1} $,\n \\item[c)] there exists an integer $k$ such that the invertibility index of $g_p(\\nu) $\nis less or equal to $k$ for all $p \\in {\\mathbb N}$.\n \\end{enumerate}\n We denote by $G_{seq}(A)$ the set of all formal group-like sequences of ${\\A} $.\n\\end{enumerate}\n\\end{defn}\n\nIt is classical that group-like elements form a group.\nMore generally:\n\n\\begin{prop}\n\\label{prop:grouplike}\nLet $( {A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta , \\epsilon, S) $ be an $(\\alpha,\\beta)$-Hom-Hopf algebra.\nGroup-like elements, formal group-like elements, $p$-order group-like elements for an arbitrary $p \\in {\\mathbb N}$, and formal group-like sequences form a Hom-group. Its product is $\\mu$, the inverse of $g$ is $S(g)$, and the unit is ${\\mathds{1}}$.\n\\end{prop}\n\\begin{proof}\n Stability under $\\mu$ of group-like elements, $k$-order group-like elements and formal group-like sequences is an immediate consequence of the compatibility relation between the multiplication and the comultiplication.\n The fact that group-like elements are hom-invertible and that $S(g)$ is a hom-inverse of a group-like element $g$ follows from \\eqref{eq:antipode_condition}, which implies that there exists $k \\in { \\mathbb N}$ such that\n $$ \\alpha^k \\circ \\mu \\,( S \\otimes \\id)\\circ \\Delta (g) = \\alpha^k \\circ \\mu \\, ( \\id \\otimes S)\\circ \\Delta (g) = \\eta \\circ \\epsilon \\, (g) ,$$\n\twhich amounts the relation\n\t$$ \\alpha^k \\circ \\mu \\, (S(g), g ) = \\alpha^k \\circ \\mu \\, (g , S(g) ) = {\\mathds{1}} .$$\n The same applies to $p$-order group-like elements, by taking the previous relation modulo $\\nu^{p+1}$.\nIt then applies for formal group-like sequences, upon noticing that the assumption\n(iii), c) in Definition \\ref{def:groupLikeAndTheLikes}\nguaranties that $ S(g_p (\\nu)) $, which is a Hom-inverse of $g_p(\\nu)$, has\nan invertibility index bounded independently from $p$, hence that $ (S(g_p (\\nu)))_{p \\in {\\mathbb N}} $ is a Hom-inverse of $ (g_p (\\nu))_{p \\in {\\mathbb N}} $.\n For formal group-like elements, the proof follows the same lines, upon noticing that for every formal group-like element $g(\\nu)=\\sum_{i= 0}^\\infty g_i \\nu^i$, there is by assumption an integer $k$ such that for all $i \\in {\\mathbb N}$:\n$$ \\alpha^k \\circ \\mu ( S \\otimes \\id)\\circ \\Delta (g_i) = \\alpha^k \\circ \\mu ( \\id \\otimes S)\\circ \\Delta (g_i) = \\eta \\circ \\epsilon \\, (g_i) = \\mathds{1} ,$$\nso that $S(g(\\nu))=\\sum_{i= 0}^\\infty S(g_i) \\nu^i$ is a Hom-inverse of $g(\\nu)$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:functorgroup}\nAlso, any morphism $\\Phi$ of $(\\alpha,\\beta)$-Hom-Hopf algebra induces in an obvious manner a morphism $\\underline{\\Phi}$ of Hom-groups between their respective group-like elements, formal group-like elements, $p$-order group-like elements for an arbitrary $p \\in {\\mathbb N}$, and formal group-like sequences.\nAssigning to an $(\\alpha,\\beta)$-Hom-Hopf algebra any of the previous types of Hom-groups, one obtains therefore a functor.\nWe call $G_{seq}$ the functor which associates to an $(\\alpha,id)$-Hom-Hopf algebra its Hom-group of formal group-like sequences.\n\\end{rem}\n\\section{Weighted trees and universal enveloping algebras as Hom-Hopf algebras}\n\\label{univ_envel_algebra}\n\nDonald Yau \\cite{Yau08,Yau4} associated to any Hom-Lie algebra $(\\gg, \\brr{\\, , \\, }, \\alpha)$ a Hom-associative algebra $ (U\\gg,\\mu,\\alpha_F)$, that he called the universal enveloping algebra of $\\gg$ and proved to be a Hom-bialgebra. His construction went through two steps: he first associated to any vector space $E$, equipped with a linear map $\\alpha: E \\to E$, the \\textbf{free Hom-nonassociative algebra} $(F_{HNAs}(E), \\mu_F, \\alpha_F)$,\nthen, he considered the quotient of this algebra through the ideal $I^{\\infty}=\\bigcup_{n \\geq 1} I^n$ where $I^1$ is the two-sided ideal\n$$\nI^1= \\langle {\\rm Im}(\\mu_F \\circ (\\mu_F \\otimes \\alpha_F - \\alpha_F \\otimes \\mu_F)); [x,y] -(xy-yx) \\mbox{ for } x,y \\in \\gg \\rangle, $$\nand $(I^n)_{n \\in {\\mathbb N}} $ is given by the recursion relation $ I^{n+1}= \\langle I^n \\cup \\alpha(I^n)\\rangle $. He did not show that the henceforth obtained Hom-bialgebra comes with an antipode.\n\nIn the multiplicative case, a more direct construction exists, that we now present. We need first to introduce some generalities about weighted trees, and to define the free Hom-associative multiplicative algebra.\nMoreover, we will see that it carries a structure of Hom-Hopf algebra.\n\n\\subsection{Weighted trees as the free Hom-associative algebra with $1$-generator}\n\nA planar tree is an oriented graph drawn on a plane with only one root. It is called binary when any vertex is trivalent, i.e., one root and two leaves. Usually we draw the root at the bottom of the tree and the leaves are drawn at the top of it:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.2, yscale=0.2]\n\n\\draw[line width=1pt] (0,-1) -- (0,2) -- (4,6);\n\\draw[line width=1pt] (3,5) -- (2,6);\n\\draw[line width=1pt] (0,2) -- (-4,6);\n\\draw[line width=1pt] (-2,4) -- (0,6);\n\\draw[line width=1pt] (-1,5) -- (-2,6);\n\n\n\\draw (-2,0) node {root};\n\\draw (-7,6) node {leaves};\n\n\\end{tikzpicture}\n\\end{center}\n\n\nFor any natural number $n\\geq 1$, let $T_n$ denote the set of planar binary trees with $n$ leaves and one root. For $n=1$, $T_1$ admits only one element, namely the unique tree with one leaf and the root. Below are the sets $T_n$ for $n=1,2,3,4$:\n$$\nT_1=\\left\\{ \\mbox{\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\end{tikzpicture}} \\: \\right\\}, \\quad T_2= \\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n} \\right\\} , \\quad T_3=\\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\end{tikzpicture}\n} \\right\\} $$\n$$ T_4=\\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n},\\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (-0.3,1.3) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (-0.3,1.3) -- (0.2,2);\n\\draw[line width=1pt] (-0.07,1.6) -- (-0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}\\right\\}.\n$$\nAn element $\\varphi \\in T_n$ shall be called an $n$-tree for short. When necessary we label the leaves of an $n$-tree by $1,2,3, \\dots, n$ from left to right.\n\nFor $\\varphi \\in T_n$ and $\\psi \\in T_m$ be a pair of trees, the $(n+m)$-tree $\\varphi \\vee \\psi$, called the \\emph{grafting of $\\varphi$ and $\\psi$}, is obtained by joining the roots of $\\varphi $ and $\\psi$ to create a new root. For instance,\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\draw (2,1) node {$\\vee$};\n\\end{tikzpicture}\n \\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (2,1) node {$=$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.3, yscale=0.3]\n\\draw[line width=1pt] (1,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (1,0) -- (1,-1);\n\\draw[line width=1pt] (1,0) -- (3,2);\n\\draw[line width=1pt] (2.5,1.5) -- (2,2);\n\\end{tikzpicture}\n\\end{center}\nNote that grafting is neither an associative nor a commutative operation. For any tree $\\varphi \\in T_n$, there are unique integers $p$ and $q$ with $p+q=n$ and trees $\\varphi_1 \\in T_p$ and $\\varphi_2 \\in T_q$ such that $\\varphi= \\varphi_1 \\vee \\varphi_2$. It is clear that any tree in $T_n$ can be obtained from \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1);\n\\end{tikzpicture} , the 1-tree, by sucessive graftings.\n\n Yau's construction of $U\\gg $ used weighted trees. In the sequel, since we are dealing with multiplicative case, it suffices to work with leaf weighted trees:\n\n\\begin{defn}\nA \\textbf{leaf weighted $n$-tree} is a pair $(\\varphi, a)$ where:\n\\begin{itemize}\n\\item $\\varphi \\in T_n$ is a $n$-tree,\n\\item $a$ is an $n$-tuple $(a_1, a_2, \\dots,a_n) \\in {\\mathbb N}^n$ of non-negative integers.\n\\end{itemize}\nWe call the tree $\\varphi$ the underlying tree of the leaf weighted $n$-tree $(\\varphi,a)$ while,\nfor all $ i=1 \\dots,n$, the integer $a_i$ shall be referred to as the \\textbf{weight} of the leaf $i$.\n\\end{defn}\n\n\nWe will indeed barely use the notation $ (\\varphi,a) $ at all, and find more convenient to picture a leaf weighted $n$-tree $(\\varphi, a_1,a_2, \\dots, a_n)$ by drawing the tree $\\varphi$ and putting the weight $a_i$ next to each leaf. For example, here are two leaf weighted $3$-trees:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$2$};\n\\draw (2,4.7) node {$1$};\n\\end{tikzpicture}\\hspace*{2cm}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (-2,4);\n\\draw[line width=1pt] (0,2) -- (2,4);\n\\draw[line width=1pt] (1,3) -- (0,4);\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$1$};\n\\draw (2,4.7) node {$0$};\n\\end{tikzpicture}\n\\end{center}\n\nThe grafting operation extends to leaf weighted $n$-trees. For example:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$2$};\n\\draw (2,4.7) node {$1$};\n\\draw (3,2) node {$\\vee$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (-2,4);\n\\draw[line width=1pt] (0,2) -- (2,4);\n\\draw[line width=1pt] (1,3) -- (0,4);\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$1$};\n\\draw (2,4.7) node {$0$};\n\\draw (3,2) node {$=$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.4) node {$0$};\n\\draw (-1.5,4.4) node {$2$};\n\\draw (-0.5,4.4) node {$1$};\n\\draw (0.5,4.4) node {$0$};\n\\draw (1.5,4.4) node {$1$};\n\\draw (2.5,4.4) node {$0$};\n\\end{tikzpicture}\n\\end{center}\n\nFor all $n \\geq 1$, we let $B_n$ denote the set of leaf weighted $n$-trees. Let $B$ denote the union over $n \\in {\\mathbb N}$ of the sets $B_n$ together with an element that we call the unit and denote by $\\mathds{1}$. Note that the element $\\mathds{1}$ is different from the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$. We then consider the free vector space ${\\mathds{T}}$ generated by the set~$B$.\n\nWe define on ${\\mathds{T}}$ two natural linear maps:\n\\begin{enumerate}\n \\item[(i)] $\\alpha:{\\mathds{T}} \\to {\\mathds{T}} $ sending $\\mathds{1}$ to $\\mathds{1}$, i.e. $\\alpha(\\mathds{1})=\\mathds{1}$ and sending a leaf weighted $n$-tree to the leaf weighted $n$-tree obtained adding $+1$ to all the weights of the leaves, i.e. $\\alpha((\\varphi, a_1,a_2, \\dots, a_n))=((\\varphi, a_1+1,a_2+1, \\dots, a_n+1))$;\n \\item[(ii)] a product $ \\vee $ that, for any pair of leaf weighted trees, is just the grafting of these trees and such that for any weighted tree $\\varphi $\n $$\n \\varphi \\vee \\mathds{1} = \\mathds{1} \\vee \\varphi = \\alpha (\\varphi)\n $$\nand $\\mathds{1}\\vee \\mathds{1} = \\mathds{1}$.\n\\end{enumerate}\n\n\nFor example\n$$\\alpha\\left( \\mbox{\\begin{tikzpicture}[xscale=0.2, yscale=0.2,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\n\\draw (-2,4.8) node {$0$};\n\\draw (0,4.8) node {$2$};\n\\draw (2,4.8) node {$1$};\n\\end{tikzpicture}} \\right)= \\mbox{ \\begin{tikzpicture}[xscale=0.2, yscale=0.2,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\\draw (-2,4.8) node {$1$};\n\\draw (0,4.8) node {$3$};\n\\draw (2,4.8) node {$2$};\n\\end{tikzpicture}}\n$$\n\nThe proof of the following lemma is trivial:\n\n\\begin{lem} \\label{alphavee}\nThe map $\\alpha$ defined in item (i) above is a morphism for the grafting of trees, that is\n$$\n\\alpha(\\varphi \\vee \\psi)= \\alpha(\\varphi)\\vee \\alpha(\\psi)\n$$\nfor any $\\varphi, \\psi \\in {\\mathds{T}}$.\n\\end{lem}\n\n\n\nWe now define an important operation which consists in eliminating leaves while changing weights of the remaining ones:\n\n\\begin{defn}\nLet $\\varphi \\in B_n$ and $I \\subset \\{ 1,2, \\dots, n \\}$, we will denote by $\\varphi_I$ the tree obtained by replacing\nall the leaves in $\\{ 1,2, \\dots, n \\} \\backslash I$ by $\\mathds{1}$. In particular, $\\varphi_{\\emptyset}=\\mathds{1}$ and $\\varphi_{\\{ 1,2, \\dots, n \\}}= \\varphi$.\n\\end{defn}\n\nAs an example, if $\\varphi$ is the tree \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.5,4.5) node {$1$};\n\\draw (2.5,4.5) node {$2$};\n\\end{tikzpicture} and $I=\\{3,5,6\\}$, then\n\n$$\\varphi_I = \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.5) node {$\\mathds{1}$};\n\\draw (-1.5,4.5) node {$\\mathds{1}$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$\\mathds{1}$};\n\\draw (1.5,4.5) node {$1$};\n\\draw (2.5,4.5) node {$2$};\n\\end{tikzpicture} =\n\\begin{tikzpicture}[xscale=0.8, yscale=0.8,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.7) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.2) node {$\\mathds{1}$};\n\\draw (-0.2,2.25) node {$0$};\n\\draw (0.2,2.25) node {$2$};\n\\draw (1,2.25) node {$3$};\n\\end{tikzpicture} =\n\\begin{tikzpicture}[xscale=0.8, yscale=0.8,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.7) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw (-1,2.2) node {$1$};\n\\draw (0.2,2.25) node {$2$};\n\\draw (1,2.25) node {$3$};\n\\end{tikzpicture}\n$$\n\n\n\\\n\n\nWe then define a coproduct $\\Delta: {\\mathds{T}} \\longrightarrow {\\mathds{T}} \\otimes {\\mathds{T}} $ by\n\\begin{equation}\\label{def:coproduct}\n\\Delta \\varphi= \\sum_{\\substack{I\\cup J= \\{1 , \\dots, n\\} \\\\ I \\cap J= \\emptyset}} \\varphi_I \\otimes \\varphi_J\n\\end{equation}\nfor a leaf weighted $n$-tree $\\varphi$, extended by linearity, and $\\Delta (\\mathds{1}) = \\mathds{1} \\otimes \\mathds{1} $.\n\n\\begin{lem}\\label{lem:coproduct-properties}\nThis coproduct satisfies the following:\n\\begin{enumerate}\n \\item[(i)] $(\\Delta \\otimes id) \\circ \\Delta = ( id \\otimes \\Delta) \\circ \\Delta$, i.e., $\\Delta$ is coassociative.\n \\item[(ii)] $\\sigma_{12} \\circ \\Delta = \\Delta$, i.e., $\\Delta$ is cocommutative (with $\\sigma_{12}: A \\otimes B \\longrightarrow B \\otimes A$ being the twist map defined by $\\sigma_{12}(a \\otimes b)= b \\otimes a$.)\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n(i) Using Definition \\ref{def:coproduct} of $\\Delta$, we have\n$$\n(\\Delta \\otimes id) \\circ \\Delta \\varphi= \\sum_{\\substack{I \\cup J \\cup K= \\{1,\\dots, n\\} \\\\ I \\cap J= \\emptyset \\\\ I \\cap K = \\emptyset \\\\ J \\cap K = \\emptyset}} \\varphi_I \\otimes \\varphi_J \\otimes \\varphi_K =( id \\otimes \\Delta) \\circ \\Delta \\varphi\n$$\nfor any $\\varphi \\in B_n$.\n\n(ii) Trivial in view of (\\ref{def:coproduct}).\n\\end{proof}\n\nWe also define a counit $\\epsilon: {\\mathds{T}} \\longrightarrow \\mathbb K$ by $\\epsilon(\\mathds{1})= 1$ and $\\epsilon( \\varphi)=0$, for any $\\varphi \\in B_n$, using then linearity. Using the above Lemma \\ref{lem:coproduct-properties}, it is easy to prove that:\n\n\\begin{prop}\\label{prop:coalgebra}\nThe triple $({\\mathds{T}}, \\Delta,\\epsilon)$ is a co-unital coassociative cocommutative coalgebra.\n\\end{prop}\n\nThis co-unital coassociative cocommutative coalgebra is compatible with the product $\\vee$ in the following sense:\n\n\\begin{lem} \\label{Deltavee}\nFor all $\\varphi, \\psi \\in {\\mathds{T}} $ the compatibility relation\n\\begin{equation}\\label{compatibility}\n\\Delta( \\varphi \\vee \\psi)= \\Delta(\\varphi) \\vee \\Delta(\\psi),\n\\end{equation}\nholds with the understanding that the right hand side of (\\ref{compatibility}) is equipped with the natural algebra structure on the tensor algebra.\n\\end{lem}\n\n\\begin{proof}\nFor $I$ a subset of $\\{1, \\dots, n\\}$, let us denote by $I^c$ the complement set.\nFor all $\\varphi \\in B_n$ and $\\psi \\in B_m$, equation (\\ref{def:coproduct}) gives\n$$\n\\Delta( \\varphi \\vee \\psi) = \\sum_{I \\subseteq \\{1, \\dots, n+m\\}} (\\varphi \\vee \\psi)_I \\otimes (\\varphi \\vee \\psi)_{I^c}.\n$$\nLet $I_n= I \\cap \\{1,\\dots,n\\}$ and $I_m = I \\cap \\{n+1, \\dots, n+m\\}$, and define $I_n^c$ and $I_m^c$ to be the complements of $I_n $ and $ I_m$ in $\\{1,\\dots,n\\}$\nand $ \\{n+1, \\dots, n+m\\}$, respectively. A direct computation gives:\n\\begin{align}\n \\Delta( \\varphi \\vee \\psi)& = \\sum_{\\substack{I_n \\subseteq \\{1, \\dots, n\\} \\\\ I_m \\subseteq \\{n+1, \\dots, n+m\\}}} (\\varphi_{I_n} \\vee \\psi_{I_m}) \\otimes (\\varphi_{I_n^c} \\vee \\psi_{I_m^c}) \\nonumber \\\\\n& = \\sum_{\\substack{I_n \\subseteq \\{1, \\dots, n\\} \\\\ I_m \\subseteq \\{1, \\dots, m\\}}} (\\varphi_{I_n} \\otimes \\varphi_{I_n^c}) \\vee (\\psi_{I_m} \\otimes \\psi_{I_m^c}) \\nonumber \\\\\n& = \\left(\\sum_{I_n \\subseteq \\{1, \\dots, n\\}} \\varphi_{I_n} \\otimes \\varphi_{I_n^c}\\right) \\vee \\left(\\sum_{ I_m \\subseteq \\{1, \\dots, m\\}}\\psi_{I_m} \\otimes \\psi_{I_m^c} \\right) \\nonumber \\\\\n& = \\Delta(\\varphi) \\vee \\Delta(\\psi). \\nonumber\n\\end{align}\nIf $ \\varphi$ or $\\psi $ are equal to $\\mathds{1}$, (\\ref{compatibility}) is equivalent to:\n\\begin{equation}\n\\label{Deltaal}\n\\Delta \\circ \\alpha= (\\alpha \\otimes \\alpha) \\circ \\Delta,\n\\end{equation}\nwhich follows directly from (\\ref{def:coproduct}). This completes the proof.\n\\end{proof}\n\nBecause the map $\\vee$ is not associative the set $({\\mathds{T}}, \\vee, \\mathds{1}, \\Delta, \\epsilon)$ is not a bialgebra. Let us consider now the bilateral ideal ${\\mathcal I}$, with respect to $\\vee$, generated by the following elements:\n $$\n (\\phi \\vee \\psi) \\vee \\alpha (\\chi) -\\alpha (\\phi) \\vee (\\psi \\vee \\chi)\n $$\nwhere $\\phi,\\psi,\\chi $ are either arbitrary leaf weighted trees or the unit $\\mathds{1}$.\nNotice that in case $\\phi, \\psi$ or $\\chi$ are equal to $\\mathds{1}$, then $c(\\phi,\\psi,\\chi)=0$, using Lemma \\ref{alphavee}.\nWe call the quotient $\\mathds{T}\/{\\mathcal I}$, equipped with its structural maps $\\vee, \\alpha$ and the unit $\\mathds{1}$,\nthe \\textbf{free Hom-associative algebra with $1$-generator}.\n\n\n\\begin{rem}\\label{rem:universal}\nThe free Hom-associative algebra with $1$-generator should be considered as an algebra that encodes all the possible operations\nthat can be described purely in terms of the Hom-associative product and $\\alpha$. These operations can then\nbe applied to an arbitrary Hom-associative algebra $\\A$.\n\\end{rem}\n\n\n\\begin{prop}\nThe coproduct $\\Delta$ and the counit $\\epsilon$ of $\\mathds{T}$ go to the quotient, and endow\nthe free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ with a Hom-bialgebra\nstructure.\n\\end{prop}\n\nWe first prove an obvious lemma:\n\n\\begin{lem}\\label{lem:coideal1}\nLet $e,f,g,h$ be elements in $ {\\mathds{T}}$.\nIf $e - f \\in {\\mathcal I}$ and $g-h \\in {\\mathcal I}$, then $e \\otimes g - f \\otimes h \\in {\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I}$.\n\\end{lem}\n\\begin{proof}\nWe just have to use the algebraic identity $e \\otimes g - f \\otimes h= (e-f) \\otimes g + f \\otimes (g-h)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:coideal2}\nThe ideal ${\\mathcal I}$ is a coideal of $({\\mathds{T}}, \\Delta,\\epsilon)$.\n\\end{lem}\n\\begin{proof}\nWe have to check that:\n\\begin{itemize}\n \\item $\\epsilon({\\mathcal I})=0$.\n \\item $\\Delta$ maps ${\\mathcal I} $ to ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I}$.\n \\end{itemize}\n The first point is trivial by definition of $\\epsilon$.\n For the second point,\nsince $\\Delta$ and $\\vee$ are compatible (Lemma \\ref{Deltavee}) it suffices indeed to show that generating elements of ${\\mathcal I} $, i.e.\nelements of the form $c(\\phi,\\psi,\\chi)=( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) -\\alpha (\\phi) \\vee (\\psi \\vee \\chi) $,\n are mapped to ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I} $.\n\n\n\nFor arbitrary $\\phi \\in B_n$, $\\psi \\in B_m$ and $\\chi \\in B_k$, we have on the one hand:\n\\begin{align}\n \\Delta\\left( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) \\right)\n & = \\left(\\Delta(\\phi) \\vee \\Delta(\\psi) \\right) \\vee \\Delta\\left( \\alpha (\\chi) \\right), \\quad \\mbox{by Lemma \\ref{Deltavee}} \\nonumber \\\\\n &= \\left(\\Delta(\\phi) \\vee \\Delta(\\psi) \\right) \\vee \\alpha ^{\\otimes^2}\\left( \\Delta (\\chi) \\right), \\quad \\mbox{by equation (\\ref{Deltaal})} \\nonumber \\\\\n& = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\left( \\phi_I \\otimes \\phi_{I^c} \\right) \\vee \\left(\\psi_J \\otimes \\psi_{J^c} \\right) \\right) \\vee \\left( \\alpha (\\chi_K) \\otimes \\alpha (\\chi_{K^c})\\right) \\nonumber \\\\\n& = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\left( \\phi_I \\vee \\psi_J \\right) \\vee \\alpha (\\chi_K) \\right) \\otimes \\left( \\left( \\phi_{I^c} \\vee \\psi_{J^c} \\right) \\vee \\alpha (\\chi_{K^c})\\right) \\nonumber,\n\\end{align}\nwhile on the other hand, we have\n\\begin{align}\n \\Delta\\left( \\alpha(\\phi) \\vee (\\psi \\vee \\chi) \\right) = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\alpha( \\phi_I) \\vee \\left( \\psi_J \\vee \\chi_K \\right)\\right) \\otimes \\left( \\alpha(\\phi_{I^c}) \\vee \\left( \\psi_{J^c} \\vee \\chi_{K^c} \\right)\\right). \\nonumber\n\\end{align}\nApplying Lemma \\ref{lem:coideal1} to $e= \\left( \\phi_I \\vee \\psi_J \\right)\\vee \\alpha (\\chi_K) , f=\\left( \\phi_{I^c} \\vee \\psi_{J^c} \\right) \\vee \\alpha (\\chi_{K^c}),\ng= \\alpha( \\phi_I) \\vee \\left( \\psi_J \\vee \\chi_K \\right), h= \\alpha(\\phi_{I^c}) \\vee \\left( \\psi_{J^c} \\vee \\chi_{K^c}\\right)$,\nwe see that the difference between $ \\Delta\\left( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) \\right)$ and $ \\Delta\\left( \\alpha(\\phi) \\vee (\\psi \\vee \\chi) \\right) $\nis an element in ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I} $, which completes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of the proposition]\nBy definition of the ideal ${\\mathcal I}$, the quotient space $\\mathds{T}\/{\\mathcal I}$, is a Hom-associative algebra\nwhen equipped with $\\vee$, and ${\\mathds{1}} $ is a unit. From Proposition \\ref{prop:coalgebra}, it also follows that $({\\mathds{T}}\/{\\mathcal I}, \\Delta,\\epsilon)$\nis a coassociative algebra with counit $\\epsilon$. According to Lemma \\ref{Deltavee}, these induced structures are compatible.\n\\end{proof}\n\nWe now intend to define an antipode on the free Hom-associative algebra with $1$-generator. We first define $S: {\\mathds{T}} \\longrightarrow {\\mathds{T}}$ by:\n\\begin{itemize}\n \\item $S(\\mathds{1})=\\mathds{1} $;\n \\item $S(|, a_1)= - (|, a_1)$, or $S\\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture}\\right) = - \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5);\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture}$, where $a_1$ is a non-negative integer;\n \\item $S$ is an antimorphism of $({\\mathds{T}},\\vee)$, i.e., $S(\\varphi \\vee \\psi)= S(\\psi) \\vee S(\\varphi)$, for any $\\varphi, \\psi \\in B$.\n\\end{itemize}\n\n\nExplicitly, $S$ maps a tree with $n$-leaves to $(-1)^n$ times the image of that tree through a vertical symmetry applied at each node, for example:\n$$\nS\\left(\n\\begin{tikzpicture}[xscale=0.35, yscale=0.35,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.75,4.5) node {$2$};\n\\end{tikzpicture}\n \\right)= (-1)^5\n \\begin{tikzpicture}[xscale=0.35, yscale=0.35,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (-1.75,4) ;\n\\draw[line width=1pt] (-1.25,3.25) -- (-0.5,4);\n\n\n\\draw (2.5,4.5) node {$2$};\n\\draw (1.5,4.5) node {$4$};\n\\draw (0.5,4.5) node {$0$};\n\\draw (-0.5,4.5) node {$3$};\n\\draw (-1.75,4.5) node {$2$};\n\\end{tikzpicture}\n$$\n\n\nThe map $S$ clearly goes to the quotient and induces an endomorphism of the free Hom-associative algebra with $1$-generator. The main result of this section is:\n\n\\begin{thm}\\label{thm:HopfAlgOnTrees}\n The free Hom-associative algebra with $1$-generator is an $(\\alpha,id )$-Hom-Hopf algebra\n (in the sense of Definition \\ref{def:hom-Hopf-algebra_in_our_sense}) when equipped with the antipode $S$.\n\\end{thm}\n\\begin{proof}\nIt is obvious that $S$ commutes with $\\alpha$ and $S({\\mathcal I}) \\subset {\\mathcal I}$, so $S$ goes to the quotient.\nThe only non-immediate result is that $S$ satisfies (\\ref{eq:antipode_condition}).\nThe result is true for $\\mathds{1}$ and for any element in $B_1$, i.e., of the form \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture} where $a_1$ is a non-negative integer.\nLet us prove that if the result holds true for leaf weighted trees $\\varphi,\\psi $ it also holds true for $\\varphi \\vee \\psi $. Let $k$ be be the non-negative number such that\n$\\alpha^k \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi)=0$, then\n\\begin{align}\n & \\alpha^{k+1} \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi \\vee \\psi) = \\nonumber \\\\\n & \\qquad = \\sum_{J,I} \\alpha^{k+1} \\circ \\vee \\circ (S \\otimes id) (\\varphi_J \\vee \\psi_I) \\otimes (\\varphi_{J^c} \\vee \\psi_{I c}) \\nonumber \\\\\n & \\qquad = \\sum_{J,I} \\alpha^{k+1}\\left( (S(\\psi_I) \\vee S(\\varphi_J)) \\vee (\\varphi_{J^c} \\vee \\psi_{I c})\\right) \\nonumber \\\\\n& \\qquad = \\sum_{J,I} \\alpha^{k}\\left( \\alpha^2(S(\\psi_I)) \\vee \\left( \\alpha (S(\\varphi_J)) \\vee (\\varphi_{J^c} \\vee \\psi_{I c}) \\right) \\right), \\mbox{ using Hom-associativity} \\nonumber \\\\\n& \\qquad = \\sum_{J,I} \\alpha^{k} (\\alpha^2(S(\\psi_I)) \\vee \\left( ( S(\\varphi_J) \\vee \\varphi_{J^c}) \\vee \\alpha(\\psi_{I c}) \\right), \\mbox{ using again Hom-associativity} \\nonumber \\\\\n& \\qquad = 0. \\nonumber\n\\end{align}\nAn induction step completes the proof.\n\\end{proof}\n\nThe antipode $S$ is an inverse in the usual sense for quite a few trees.\nBy inverse in the usual sense we mean that we can take $k=0$ in (\\ref{eq:antipode_condition}).\nLet us call \\textbf{ferns} weighted trees whose underlying tree is such that each node is\nrelated to a leaf. For instance, the following trees are ferns:\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture} and\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture} while\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture}, is not a fern.\n\n\n\nWe still call \\textbf{space of ferns} the subspace of the free vector space $\\mathds{T} $ generated by ferns and the subspace of the free Hom-associative algebra with $1$-generator which is the image of ferns in $\\mathds{T} $ through the canonical projection from $ \\mathds{T} $\nto the free Hom-associative algebra with $1$-generator.\n\nLet us investigate the subspace of the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ made of all elements with invertibility index $0$, i.e. the subspace of all elements $g \\in \\mathds{T}\/{\\mathcal I}$ such that the following relation holds\n\\begin{equation} \\label{eq:antipodeK0}\n \\vee \\circ (S \\otimes \\id) \\circ \\Delta (g)= \\vee \\circ( \\id \\otimes S) \\circ \\Delta (g)= \\eta \\circ \\epsilon \\, (g).\n\\end{equation}\n\n\\begin{prop}\\label{prop:invert-index-ferns}\nThe subspace of the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ made of all elements with invertibility index $0$ is stable under left or right multiplication under elements in $B_1$.\nIt contains the space of ferns, as well as the spaces $B_i, i=1,2,3,4$.\n\\end{prop}\n\\begin{proof}\nLet $\\varphi$ be an element in $\\mathds{T}$ where (\\ref{eq:antipodeK0}) holds, then it also holds for any tree of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} $\\vee \\varphi$ (in this case we need to use the Hom-associativity of $\\vee$):\n\n\\begin{align}\n \\vee \\circ (S \\otimes id ) \\circ \\Delta (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} \\vee \\varphi) & = \\vee \\circ (S \\otimes id ) \\circ \\left( \\Delta (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} ) \\vee \\Delta(\\varphi)\\right) \\nonumber \\\\\n& = \\vee \\circ (S \\otimes id ) \\circ \\left( (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} \\otimes \\mathds{1} + \\mathds{1} \\otimes \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} ) \\vee (\\varphi_J \\otimes \\varphi_{J^c})\\right) \\nonumber \\\\\n& = \\vee \\circ (S \\otimes id ) \\circ \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {$\\varphi_J$};\n\\draw (-1,2.3) node {$\\mathds{1}$};\n\\end{tikzpicture} \\otimes \\alpha(\\varphi_{J^c}) + \\alpha(\\varphi_{J}) \\otimes \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {$\\varphi_{J^c}$};\n\\draw (-1,2.3) node {$\\mathds{1}$};\n\\end{tikzpicture} \\right) \\nonumber\n\\end{align}\nwhich proves the claim.\nIt turns out that this also proves that relations\n $ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi)= \\mu \\circ( id \\otimes S) \\circ \\Delta (\\varphi)=0$\nhold for all ferns, since any fern in $B_{n+1}$ is obtained out of\na fern in $B_n$ by either left or right grafting with an element in $B_1$.\nIn particular, this relation also holds for all elements in $B_2,B_3$, since those spaces are included\nin the space of ferns.\nTo check it for all elements in $B_4$, it suffices to check it in the unique element which is not a fern, i.e.\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture},\n which is a direct computation.\n\\end{proof}\n\nWe give an example of an element in the free Hom-associative algebra with $1$-generator for which the antipode does not satisfy Relation (\\ref{eq:antipodeK0}):\n\\begin{center}\n \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] ((0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.75,3.7) -- (-2,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (-0.75,3.7) -- (-1,4);\n\\draw[line width=1pt] (1.35,2.8) -- (0.25,4);\n\\draw[line width=1pt] (0.7,3.5) -- (1.2,4);\n\\draw[line width=1pt] (0.95,3.7) -- (0.7,4);\n\\draw[line width=1pt] (2.25,3.7) -- (2,4);\n \\end{tikzpicture}\n \\end{center}\n\nProof: Left to the reader.\n\n\\subsection{Universal enveloping Lie algebra of a Hom-Lie algebra}\n\nLet $(\\gg, \\brr{\\, , \\, }, \\alpha)$ be a multiplicative Hom-Lie algebra. Let us apply the so-called Schur functor \\cite{Loday} to ${\\mathds{T}} $ and $ \\gg$. We define ${\\mathds{T}}^\\gg $ to be the vector space:\n$$\n {\\mathds{1}} \\oplus \\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n} .\n $$\nElements of $B_n \\otimes \\gg^{\\otimes n}$ shall be pictured for every $ \\phi \\in B_n$, $x_1, \\dots,x_n \\in \\gg$,\nby inserting, for all $i=1, \\dots,n$, the element $ x_i$ at the top of the leaf with label $x_i$:\n $$ \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.75,4.5) node {$1$};\n\\draw (-2.5,5.2) node {$x_1$};\n\\draw (-1.5,5.2) node {$x_2$};\n\\draw (-0.5,5.2) node {$x_3$};\n\\draw (0.5,5.2) node {$x_4$};\n\\draw (1.75,5.2) node {$x_5$};\n\\end{tikzpicture}$$\nFrom now, we only write down the weight of a given leaf of an element in $B_n \\otimes \\gg^{\\otimes n}$ when it is not equal to zero.\nSo \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (0,2.5) node {$ 2$};\n\\end{tikzpicture}\nis a short hand for\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.5) node {$0$};\n\\draw (0,2.5) node {$ 2$};\n\\draw (-1,2.5) node {$0$};\n\\end{tikzpicture}.\n\nThe operations $ \\vee,S,\\Delta,\\epsilon, \\alpha$ defined on $ {\\mathds{T}} $ have natural extensions\nto ${\\mathds{T}}^\\gg $, that we denote by the same symbols. The extension $\\Delta$ is still a coproduct with counit $\\epsilon$, which is still compatible with $\\vee $.\nMoreover, the subspace $ {\\mathcal I}^\\gg = \\oplus_{n \\geq 3} {\\mathcal I}_n \\otimes \\gg^{\\otimes n} $,\n(with $ {\\mathcal I}_n $ being the subspace of ${\\mathcal I} \\cap B_n $) is an ideal for $\\vee$\n and a coideal for $\\Delta$. It follows directly from Theorem \\ref{thm:HopfAlgOnTrees} that the sextuple\n$({\\mathds{T}}^\\gg\/{\\mathcal I}^\\gg,\\vee,\\Delta \\circ \\alpha,S,\\mathds{1},\\epsilon) $ is an $(\\alpha,id)$-Hom-Hopf algebra.\nWe call this Hom-Hopf algebra the \\textbf{free Hom-associative algebra with generators in~$\\gg$}.\n\nWe now consider the quotient of the latter Hom-Hopf algebra by the ideal $ {\\mathcal J}^\\gg$ of $({\\mathds{T}}^\\gg\/{\\mathcal I}^\\gg,\\vee,\\mathds{1})$ generated by:\n\\begin{enumerate}\n \\item[(i)] elements of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture},\n(recall that for all $ y \\in {\\mathfrak g} $, \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$y$};\n\\end{tikzpicture} is a short hand for \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.95) node {$0$};\n\\draw (0,2.75) node {$y$};\n\\end{tikzpicture} )\n \\item[(ii)] elements of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture}.\n\\end{enumerate}\n\nWe first establish the following result.\n\n\\begin{prop}\nThe ideal $ {\\mathcal J}^\\gg$ is a Hom-Hopf ideal of\nthe free Hom-associative algebra with generators in $\\gg$.\n\\end{prop}\n\\begin{proof}\nAgain, it suffices to show that the structural maps go to the quotient with respect to ${\\mathcal J}^\\gg $.\nSince $\\Delta $ and $\\vee $ are compatible in the sense of equation (\\ref{compatibility}), to show that $\\Delta $ goes to the quotient suffices to show that\nfor all $ x,y \\in \\gg$, both\n$ \\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $ and $\\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) $\nare elements in\n${\\mathcal J}^\\gg \\otimes {\\mathds{T}}^\\gg\/ {\\mathcal{I}}^\\gg + {\\mathds{T}}^\\gg\/ {\\mathcal{I}}^\\gg\\otimes {\\mathcal J}^\\gg$.\nThis follows from the following two computations, obtained by using directly the definition (\\ref{def:coproduct}) of the coproduct :\n$$\\Delta\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right)= \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) \\otimes \\mathds{1} + \\mathds{1} \\otimes \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $$\nand\n$$\\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right)= \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) \\otimes \\mathds{1} + \\mathds{1} \\otimes \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right).$$\nThe antipode $S$ being an antimorphism of $\\vee $, it suffices to check that it preserves the set of generators of ${\\mathcal J}^\\gg $ to state that preserves ${\\mathcal J}^\\gg$.\nThis simply follows from the definition of $S$, since:\n $$ S\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) = -\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $$\nand\\footnote{Notice that this is the first time that we are using the skew-symmetry of the bracket $[.,.]$ on ${\\mathfrak g}$.} $S\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) = - \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) .$\n\nThis completes the proof.\n\\end{proof}\n\n\\begin{defn}\\label{def:Hom-Universal}\nThe {\\bf universal enveloping algebra of a multiplicative Hom-Lie algebra $(\\gg, \\brr{\\, , \\, }, \\alpha)$}\nis by definition the quotient of the free Hom-associative algebra with generators in $\\gg$ (which is an $(\\alpha,\\id)$-Hom-Hopf algebra) by the Hom-Hopf ideal ${\\mathcal J}^\\gg $. This quotient is itself an $(\\alpha,\\id)$-Hom-Hopf algebra that we denote by ${\\mathcal U}\\gg $, while we keep the usual symbols for its structural maps.\n\\end{defn}\n\n\\begin{ex}\n For $\\alpha=id$, Hom-Lie algebras are just Lie algebras. It is routine to check that the universal enveloping algebra of $(\\gg, \\brr{\\, , \\, }, id) $ coincides with the usual universal enveloping algebra.\n\n\n\nFor $\\alpha=0$ and $[.,.]$ an arbitrary skew-symmetric map, all weighted trees are in the ideal ${\\mathcal I}^{\\mathfrak g}$ unless the weight of each leaf is $0$ and the universal enveloping algebra is obtained by applying the Schur functor to the algebra of all binary trees, then dividing the outcome by the ideal (for grafting) generated by\n$$\n \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture}\n$$\nfor all $x,y \\in {\\mathfrak g}$. This is of course not associative (it is by construction Hom-associative with respect to a map that satisfies $\\alpha=0$ on (the image of) $\\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n}$, which is not a strong constraint). Also, the coproduct is simply:\n$ \\Delta (\\phi) = \\phi \\otimes {\\mathds{1}}+ {\\mathds{1}} \\otimes \\phi$ for all $ \\phi \\in \\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n}$.\n\\end{ex}\n\n\\begin{rem}\\label{rem:functor}\nAny Hom-Lie algebra morphism $\\psi:\\gg \\to \\gg'$ induces a natural $(\\alpha,\\id)$-Hom-Hopf algebra morphism\n${\\mathcal U}\\phi : {\\mathcal U}\\gg \\to {\\mathcal U}\\gg'$ by:\n $$ {\\mathcal U}\\phi : \\phi \\otimes (x_1 \\otimes \\dots \\otimes x_n ) \\mapsto \\phi \\otimes (\\psi(x_1) \\otimes \\dots \\otimes \\psi(x_n) ) $$\nfor all weighted $n$-tree $\\phi$ and $x_1, \\dots,x_n \\in \\gg$.\nAssociating to a Hom-Lie algebra $\\gg$ a universal envelopping algebra ${\\mathcal U}\\gg$,\none therefore obtains a functor ${\\mathcal U} $ from the category of Hom-Lie algebras to the category of\n$(\\alpha,\\id)$-Hom-Hopf algebras.\n\\end{rem}\n\n Notice that for Hom-Lie algebras constructed by composition out of a Lie algebra $\\gg $, equipped with a bracket $[.,.]_{Lie}$, through an endomorphism $\\alpha$, there are two natural Hom-Hopf algebra structures:\n\\emph{(i)} the universal enveloping algebra of the Hom-Lie algebra $ (\\gg,\\alpha \\circ [.,.],\\alpha)$\nas in Definiton \\ref{def:Hom-Universal} and \\emph{(ii)} the $(\\tilde{\\alpha},\\id)$-twist of the universal enveloping algebra (in the usual sense) $U^{Lie}(\\gg) $\n of the Lie algebra $ (\\gg, [.,.]_{Lie})$\nthrough the Hopf algebra morphism $\\tilde{\\alpha} $ associated to $\\alpha $ (this is an $(\\alpha,\\id) $-Hopf-algebra by Example \\ref{Hom-Hopf-Twist}).\n\nThese two Hom-Hopf algebra do not agree. This is easily seen when $\\alpha=0$, for\nthe product in simply $0$ for the Hom-Hopf algebra of item \\emph{(ii)} (since its product is $\\mu_{\\alpha}=\\alpha \\circ \\mu$\nwith $\\mu$ the product of $U^{Lie}(\\gg)$), while it is not zero for\nthe universal enveloping algebra of the Hom-Lie algebra $ (\\gg,\\alpha \\circ [.,.],\\alpha)$.\nFor instance, the product\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$x$};\n\\end{tikzpicture}\n$\\vee$\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$x$};\n\\end{tikzpicture}\n\\end{center}\ndoes not vanish if $ x \\in {\\mathfrak g}$ a non-zero element. Notice that the spaces of which they are defined also do not coincide.\n\n\n\n\\subsection{Primitive elements on the free Hom-associative algebra with $1$-generator}\n\nBesides the properties of primitive elements of an $(\\alpha,\\beta)$-Hom-Hopf algebra described in Remark \\ref{rmk:SeveralPoints}, we aim to discuss the case of $\\mathds{T}\/{\\mathcal I}$.\nBy construction, $\\alpha :\\mathds{T} \\to \\mathds{T}$ is injective, and it is natural to ask\nif its induced map $\\alpha$ on the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ is injective as well. The answer is negative, in view of the next proposition, which introduces a useful element in building counterexamples. It shows in particular that although all elements in $B_1$ (i.e. leaf weighted $1$-trees) are primitive for the coassociative coproduct (defined as in second item of Remark \\ref{rmk:SeveralPoints}), the transpose is not true: there are primitive elements which are not in $B_1$.\n\n\\begin{prop}\\label{prop:remarquable}\nConsider the following element in $\\mathds{T}$:\n$$\n\\begin{tikzpicture}[xscale=0.6, yscale=0.6, baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.3) node {$0$};\n\\draw (-0.2,2.3) node {$1$};\n\\draw (0.2,2.3) node {$1$};\n\\draw (1,2.3) node {$0$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.6, yscale=0.6,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.3) node {$1$};\n\\draw (-0.2,2.3) node {$0$};\n\\draw (0.45,2.3) node {$0$};\n\\draw (1,2.3) node {$0$};\n\\end{tikzpicture}.\n$$\nDenote by $u$ its class in the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$.\nThen $u \\neq 0$ but $\\alpha (u) =0$. Moreover, $u$ is a primitive element, and any element in the algebra (for $\\vee$) generated by $u $ is primitive.\n\\end{prop}\n\\begin{proof}\nIt is clear that the tree that defines $u$ is not contained in $ {\\mathcal I}$, which is therefore not equal to zero.\nAlso, $\\alpha(u)= \\begin{tikzpicture}[xscale=0.6, yscale=0.6, baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.3) node {$1$};\n\\draw (-0.2,2.3) node {$2$};\n\\draw (0.2,2.3) node {$2$};\n\\draw (1,2.3) node {$1$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.6, yscale=0.6,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.3) node {$2$};\n\\draw (-0.2,2.3) node {$1$};\n\\draw (0.45,2.3) node {$1$};\n\\draw (1,2.3) node {$1$};\n\\end{tikzpicture}=0$, using Hom-associativity. By a direct computation, $\\Delta(u)= u \\otimes \\mathds{1} + \\mathds{1} \\otimes u$, i.e. $u$ is a primitive element.\n\nAs a consequence, $ u$ is a primitive element contained in the kernel of $ \\alpha$.\nNow, for every pair $ u_1,u_2$ of primitive elements contained in the kernel of $\\alpha$,\n$u_1 \\vee u_2$ is a primitive element, as follows from equation (\\ref{compatibility}):\n \\begin{eqnarray*} \\Delta (u_1 \\vee u_2 ) &=& \\Delta (u_1 ) \\vee \\Delta (u_2) \\\\\n &=& (u_1 \\otimes \\mathds{1} + \\mathds{1} \\otimes u_1) \\otimes ( u_2 \\otimes \\mathds{1} + \\mathds{1} \\otimes u_2) \\\\\n& = & (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\\\\n& & + ( u_1 \\vee \\mathds{1} ) \\otimes ( \\mathds{1} \\vee u_2 ) +( \\mathds{1} \\vee u_2) \\otimes (u_1 \\vee \\mathds{1} ) \\\\\n& = & (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\\\\n& & + \\alpha( u_1) \\otimes \\alpha( u_2 ) + \\alpha (u_2) \\otimes \\alpha (u_1) \\\\\n & =& (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\end{eqnarray*}\nMoreover, Equation (\\ref{alphavee}) implies that any element in the algebra generated by $u $ is in the kernel of $\\alpha$. Altogether, these properties imply that the space of elements with are primitive and in the kernel of $\\alpha$\nis stable under $\\vee$. In particular, every element in the algebra generated by $u$ is primitive.\n\\end{proof}\n\n\\begin{rem}\n In view of Remark \\ref{rem:universal}, Proposition \\ref{prop:remarquable} implies that for any Hom-associative algebra $(\\A,\\vee, \\alpha)$, and any\n $x,y,z,t \\in \\A$, the element\n $$ \\alpha(t) \\vee ((x \\vee y) \\vee t) - (t\\vee \\alpha(x)) \\vee (\\alpha(y) \\vee z)$$\nis in the kernel of $\\alpha$.\n\\end{rem}\n\n\n\n\\subsection{Canonical $n$-ary operations on Hom-associative algebras}\nLet $A$ be a Hom-associative algebra.\nOn $ \\A$, making the product of $n$ elements, $n \\geq 3$ depends on the order in which the products are taken. There is however a natural manner manner to define $n$-ary operations $\\A^{\\otimes n} \\to \\A $, as we will see in the sequel. Recall that by Remark \\ref{rem:universal}, operations of $n$ elements of $A$ are encoded by leaf weighted $n$-trees.\n\nGiven a $n$-tree $ \\varphi \\in T_n $ (without weights) and an integer $n \\leq k$,\nwe define a leaf weighted $n$-tree $\\varphi[k] $ by assigning to the leaf with label $i$ the integer\n$ k - \\ell (i)$ with $ \\ell$ being the length of the branch from the root to the leaf. The length of the leaf $i$ is then the length of the path from the root to the leaf,\nwhen the tree is seen as a graph.\nFor instance, let\n$\\varphi=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\end{tikzpicture}\n$ then\n$\\varphi[7]=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\draw (-2.5,4.5) node {$3$};\n\\draw (-1.5,4.5) node {$3$};\n\\draw (-0.5,4.5) node {$4$};\n\\draw (0.5,4.5) node {$4$};\n\\draw (1.75,4.5) node {$4$};\n\\end{tikzpicture}\n$ and $\\varphi[5]=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\draw (-2.5,4.5) node {$1$};\n\\draw (-1.5,4.5) node {$1$};\n\\draw (-0.5,4.5) node {$2$};\n\\draw (0.5,4.5) node {$2$};\n\\draw (1.75,4.5) node {$2$};\n\\end{tikzpicture}\n$.\n\nLet us call \\textbf{right $n$-fern} the $n$-tree (without weights) obtained by successive graftings on the right of the $1$-tree and denote it by $F_n^r$, i.e. $$F_n^r= \\left(\\left(\\dots \\left( \\left(\\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture}\\right) \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\vee \\dots \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture}\\right)= \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (2.2,4) ;\n\\draw[line width=1pt] (-0.3,1.75) -- (1.65,4) ;\n\\draw (0.4,4) node {$\\dots$};\n\\end{tikzpicture}$$\nOf course, the right $n$-fern is a fern. Similarly we call \\textbf{left $n$-fern} the $n$-tree (without weights) obtained by successive graftings on the left of the $1$-tree and denote it by $F_n^l$, i.e. $$F_n^l= \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\dots \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\right) \\dots \\right) \\right)\n= \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (-2.2,4) ;\n\\draw[line width=1pt] (0.3,1.75) -- (-1.65,4) ;\n\\draw (-0.4,4) node {$\\dots$};\n\\end{tikzpicture}$$\n\nWe first prove a lemma:\n\n\\begin{lem}\\label{KFougeres}\n Let $k,n $ be non-negative integers with $ n \\leq k$. The identity\n $$\n F_n^l[k]= F_n^r[k]\n $$\nholds in the free Hom-associative algebra with 1-generator ${\\mathds{T}}\/{\\mathcal I} $.\n\\end{lem}\n\n\\begin{proof}\nThis follows by a finite induction using definitions of these trees and Hom-associativity.\n\\end{proof}\nThe following Lemma is straightforward.\n\\begin{lem}\\label{lemmaKtoK-1}\nLet $\\varphi_1 \\in T_p, \\varphi_2 \\in T_q $ be two trees and $k$ an integer such that $k\\geq p+q$. Then\n$$\n(\\varphi_1\\vee \\varphi_2)[k]= \\varphi_1[k-1] \\vee \\varphi_2[k-1].\n$$\n\\end{lem}\nNow, we state the main result of this section showing that the operations on Hom-associative algebras encoded by the weighted $n$-trees $\\varphi[k]$ depends only on $n$ and $k$.\n\\begin{prop}\n\\label{prop:indifference}\n Let $k,n $ be two non-negative integers with $ n \\leq k$. For any $n$-trees $\\varphi,\\psi \\in T_n$, the identity $\\varphi[k] = \\psi[k]$ holds in the free Hom-associative algebra with 1-generator ${\\mathds{T}}\/{\\mathcal I} $.\n\\end{prop}\n\\begin{proof}\nIt suffices to show that, for any tree $\\varphi \\in T_n$, we have $\\varphi[k]= F_n^r[k]$. For $n=1,2,3$, it is routine to check that this identity is true (using Hom-associativity in case $n=3$). Let us suppose now that this equality holds for any tree in $T_p$ with $p< n$. Let $\\varphi$ be a tree in $T_n$ with $n>3$, then there exist $\\varphi_1 \\in T_p, \\varphi_2 \\in T_q $ such that $\\varphi= \\varphi_1 \\vee \\varphi_2$ and $p+q=n$. By Lemma \\ref{lemmaKtoK-1} we have\n$$\n\\varphi[k]= \\varphi_1[k-1] \\vee \\varphi_2[k-1]\n$$\nand applying the hypothesis $\\varphi_1[k-1]= F^r_p[k-1]$ and $\\varphi_2[k-1]= F^r_q[k-1]=F^l_q[k-1]$, using also the Lemma \\ref{KFougeres}. If $q=1$, then $\\varphi[k]=F^r_p[k-1] \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.7) node {\\tiny $k-2$};\n\\end{tikzpicture} = F^r_n[k] $, if not, using in each step Hom-associativity, we have $\\varphi[k]=F^r_p[k-1] \\vee F^l_q[k-1] = F^r_{p+1}[k-1] \\vee F^l_{q-1}[k-1]= F^r_{p+2}[k-1] \\vee F^l_{q-2}[k-1]= \\dots=F^r_{p+q-1}[k-1] \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.7) node {\\tiny $k-2$};\n\\end{tikzpicture}= F^r_{p+q}[k] = F^r_{n}[k]$.\n\\end{proof}\n\nAs a consequence, for every non-negative integers $k,n $ with $ n \\leq k$, we denote by $\\lfloor e^n \\rfloor_k $ the element in ${\\mathds{T}}\/{\\mathcal I} $, defined by $\\varphi[k] \\in {\\mathds{T}}\/{\\mathcal I} $ for an arbitrary $n$-tree $ \\varphi \\in T_n$. It is called the \\textbf{$k$-weighted $n$-ary product}.\n\n\\begin{ex} $ \\lfloor e^1 \\rfloor_k= $\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$k-1$};\n\\end{tikzpicture} , $\\lfloor e^2 \\rfloor_k= $\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$k-2$};\n\\draw (-1,2.2) node {\\scriptsize$k-2$};\n\\end{tikzpicture} , $\\lfloor e^3 \\rfloor_k=$ \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$k-3$};\n\\draw (0,2.1) node {\\scriptsize$ k-3$};\n\\draw (-1,2.1) node {\\scriptsize$ k-2$};\n\\end{tikzpicture}.\n\\end{ex}\n\\begin{lem}\\label{lem:coproduct-explicit}\n For all non-negative integers $k,n,m $ with $ n < k$ and $ m < k$, we have\n\\begin{enumerate}\n\\item\n $ \\Delta ( \\lfloor e^n \\rfloor_k) = \\sum_{i=0}^n\n\\left(\n\\begin{matrix}\n n \\\\\ni\n\\end{matrix}\n\\right)\n \\lfloor e^i\\rfloor_k \\otimes \\lfloor e^{n-i}\\rfloor_{k} ,$\n\\item\n $ \\lfloor e^n\\rfloor_k \\vee \\lfloor e^m\\rfloor _k = \\lfloor e^{n+m}\\rfloor_{k+1} =\\alpha (\\lfloor e^{n+m}\\rfloor_k), $\n \\item $S(\\lfloor e^{n}\\rfloor_{k} )=(-1)^n\\lfloor e^{n}\\rfloor_{k}.$\n \\end{enumerate}\n\\end{lem}\n\n\\\n\nGiven a formal power series in one variable with real coefficients $ f(\\nu)= \\sum_{i \\geq 0}^\\infty a_i \\nu^i$, we denote by $ \\widehat{f}_p (\\nu)$ and call it \\textbf{ $k$-weighted realization of $f$ } the element in ${\\mathds{T}}\/{\\mathcal I}[[\\nu]]$ modulo $\\nu^{p+1}$ given by:\n\\begin{equation}\\label{ChapFnu}\n\\widehat{f}_p(\\nu) =a_0 \\mathds{1}+ \\sum_{i \\geq 1}^p a_i \\nu^i \\lfloor e^i \\rfloor_p.\n\\end{equation}\nWe call the sequence $(\\widehat{f}_p(\\nu))_{p\\in \\mathbb{N}}$ the \\textbf{ realization of $f$ } and denote it by $\\widehat{f}(\\nu)$.\\\\\n\nWe provide the following properties:\n\n\\begin{prop}\\label{Prop:proprietiesWidehat}\nGiven two formal power series in one variable with real coefficients $ f(\\nu)= \\sum_{i \\geq 1}^\\infty a_i \\nu^i$,\n$ g(\\nu)= \\sum_{i \\geq 1}^\\infty b_i \\nu^i$. We have:\n\\begin{enumerate}\n \\item $\\widehat{f}(\\nu)\\vee \\widehat{g}(\\nu)=\\alpha( \\widehat{fg}(\\nu)),$\n \\item for all $k\\in \\mathbb{N}$, $\\widehat{f}_{p+1}(\\nu)=\\alpha(\\widehat{f}_p(\\nu))$ modulo $\\nu^{p+1}$,\n \\item $S(\\widehat{f}(\\nu))=\\widehat{f}(-\\nu)$,\n\t\\item the invertibility index of $\\widehat{f}_p(\\nu)$ is equal to $0$ for all $p \\in {\\mathbb N}$,\n\\end{enumerate}\nwhere $\\alpha$, $\\vee$ and $S$ are the structure operations of ${\\mathds{T}}\/{\\mathcal I} $ extended by $\\mathbb{R}[[\\nu]]$-linearity to ${\\mathds{T}}\/{\\mathcal I} [[\\nu]]$.\n\\end{prop}\n\\begin{proof}\nThe first assertion follows from the second item of Lemma \\ref{lem:coproduct-explicit}. The second one is obtained by considering the definition \\eqref{ChapFnu} and the relation $ \\lfloor e^{i}\\rfloor_{p+1} =\\alpha (\\lfloor e^{i}\\rfloor_p)$. The third one is a consequence of the third item of Lemma \\ref{lem:coproduct-explicit}.\nLet us prove the last assertion. Proposition \\ref{prop:indifference} implies that\n$$\\widehat{f}_p(\\nu) =a_0 \\mathds{1}+ \\sum_{i \\geq 1}^p a_i \\nu^i \\lfloor e^i \\rfloor_p$$\ncan be represented by ferns. The conclusion then follows from Proposition \\ref{prop:invert-index-ferns}.\n\\end{proof}\n\n\n\n\n\n\\section{A Hom-group integrating a Hom-Lie algebra}\n\nIn this section, we aim to associate to any Hom-Lie algebra a Hom-group.\nThis construction uses the study of the universal enveloping algebra\nand elements of group-like type.\n\n\n\\subsection{Group-like elements in the free Hom-associative algebra with $1$-generator}\n\nFor $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ a formal group-like element, $g_1$ is primitive. Unlike for the free associative algebra with $1$-generator, not any primitive element of ${\\mathds{T}}\/{\\mathcal I}$ could be the first order element of a formal group-like element with $g_0={\\mathds 1}$.\n\n\\begin{prop}\\label{prop:negativeGroupLike}\nThe $(\\alpha,\\id)$-Hom-Hopf algebra ${\\mathds{T}}\/{\\mathcal I}[[\\nu]]$ does not admit formal group-like elements\nof the form $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ where $g_1$ is the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$.\n\\end{prop}\n\\begin{proof}\nAssume that $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ is a formal group element. For example, the coefficient of $\\nu^2$ in $\\Delta (g(\\nu))=g(\\nu)\\otimes g(\\nu) $ yields\n$$\\Delta (g_2)= g_1\\otimes g_1+ g_2\\otimes {\\mathds 1}+{\\mathds 1}\\otimes g_2.$$\nThis imposes that $g_2$ is in $B_2$ which is impossible, because the projection of $\\Delta (g_2)$ on $B_1\\otimes B_1$ is a linear combination of elements of the form\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$k$};\n\\end{tikzpicture}\n$\\otimes$\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$l$};\n\\end{tikzpicture}\nwith $k$ or $l$ strictly positive.\n\\end{proof}\n\\begin{rem}\nThe proof of Proposition \\ref{prop:negativeGroupLike} gives indeed that\n there is no $2$-order formal group-like elements of the form\n$g(\\nu)= {\\mathds 1}+\\sum_{i=1}^p {g_i\\nu^i}$ where $g_1$ is the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$.\n\\end{rem}\n\nLet us show that formal group-like sequence is a relevant object by showing that the free Hom-associative algebra with $1$-generator admits a $1$-parameter family of formal group-like sequences, although it admits very few\ngroup-like elements.\n\nFor all $s\\in {\\mathbb R}$, consider the realization $\\widehat{exp}(s)$ of the formal series $exp(s)=\\sum_{i=0}^\\infty{\\frac{s^i}{i!}\\nu^i}$. We call the assignment $s \\rightarrow \\widehat{exp}(s)$ the \\textbf{exponential sequence}.\n\n\nFor a better understanding of $\\widehat{exp}(s)$, we give its first terms:\n$$\n\\begin{array}{rcl}\n\\widehat{exp}_0(s)&=& {\\mathds{ 1}} \\\\ \\widehat{exp}_1(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$0$};\n\\end{tikzpicture} \\\\\n \\widehat{exp}_2(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$1$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$0$};\n\\draw (-1,2.2) node {\\scriptsize$0$};\n\\end{tikzpicture} \\\\\n\\widehat{exp}_3(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$2$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$1$};\n\\draw (-1,2.2) node {\\scriptsize$1$};\n\\end{tikzpicture}\n+\\frac{s^3 \\nu^3}{3!}\n \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$0$};\n\\draw (0,2.1) node {\\scriptsize$ 0$};\n\\draw (-1,2.1) node {\\scriptsize$ 1$};\n\\end{tikzpicture}.\n\\end{array}\n$$\nGenerally, we have\n$$\\widehat{exp}_p(s) =\\mathds{1}+ \\sum_{i = 1}^p \\frac{ s^i}{i!} \\nu^i \\lfloor e^i \\rfloor_p .\n$$\n\n\\begin{thm}\n\\label{theo:group-like_order_kappa}\n The exponential sequence $s \\rightarrow \\widehat{exp}(s)$ is valued in\n the Hom-group $G_{seq}({\\mathds T}\/{\\mathcal I}) $ of formal group-like sequence.\\\\\n Moreover,\n \\begin{enumerate}\n \\item $ \\widehat{exp}(0) $ is the unit element of the Hom-group of formal group-like sequences.\n \\item For all $s,t \\in {\\mathbb R}, k \\in {\\mathbb N}$,\n $ \\widehat{exp}(s) \\vee \\widehat{exp}(t) = \\alpha(\\widehat{exp}(s+t)) $.\n \\item $ S(\\widehat{exp}(s)) = \\widehat{exp}(-s)$ is a strict inverse, i.e.\n $$ \\widehat{exp}(s) \\vee \\widehat{exp}(-s) = \\widehat{exp}(-s) \\vee \\widehat{exp}(s) = \\mathds{1} . $$\n \\end{enumerate}\n\\end{thm}\n\\begin{proof}\nThe first item just follows from the definition of the realization of the formal series $f(\\nu)=e^{0\\nu}=1$.\nThe second item follows from the first item in Proposition \\ref{Prop:proprietiesWidehat} applied to the classical relation $e^{s \\nu} e^{t \\nu} = e^{(s+t)\\nu}$. The third item follows from the third item in Proposition \\ref{Prop:proprietiesWidehat}.\n\nIt remains to show that the exponential sequence is valued in formal group-like sequence, defined in item (iii)\nof Definition \\ref{def:groupLikeAndTheLikes}. The fourth item in Proposition \\ref{Prop:proprietiesWidehat} implies that $\\widehat{exp}_p(s)$ (i.e. the $p$-th term in the sequence $\\widehat{exp}(s)$) has invertibility index equal to $0$, so that condition c) holds.\nThe second item in Proposition \\ref{Prop:proprietiesWidehat} implies item b) in\nDefinition \\ref{def:groupLikeAndTheLikes}. We are left with the task of showing that\n$\\widehat{exp}_p(s)$ is a $p$-order group-like element. This follows from the following computation,\nwhich is done modulo $\\nu^{p+1}$:\n\\begin{eqnarray*} \\Delta (\\widehat{exp}_p(s)) &=& \\Delta(\\mathds{1})+ \\sum_{i = 1}^p \\frac{ s^i}{i!} \\nu^i \\Delta \\lfloor e^i \\rfloor_p \\\\\n&=& \\mathds{1} \\otimes \\mathds{1}+ \\sum_{i = 1}^p \\sum_{j = 1}^i \\frac{ s^i}{i!} \\nu^i \\left( \\begin{matrix}\n i \\\\\nj\n\\end{matrix}\\right) \\lfloor e^{j} \\rfloor_p \\otimes\n \\lfloor e^{i-j} \\rfloor_p \\\\\n&=& \\widehat{exp}_p(s) \\otimes \\widehat{exp}_p(s),\n\\end{eqnarray*}\nwhere the first item of Lemma \\ref{lem:coproduct-explicit} was used to go from the first to the second line.\n\\end{proof}\n\n\n\n\\subsection{Formal group-like sequences of the universal enveloping algebra}\n\nNow, we define the exponential map for a Hom-Lie algebra $(\\gg,[\\cdot,\\cdot],\\alpha)$, in order to achieve a construction of a functor from the category of Hom-Lie algebras to the category of Hom-groups.\n For all $x_1, \\dots,x_i \\in \\gg$, and $p,i \\in \\mathbb{N}$ with $ p \\geq i $,\ndefine an element in ${\\mathcal U}\\gg$ by:\n$$ \\lfloor x_1, \\dots,x_i \\rfloor_p = e^i_p \\otimes (x_1 \\otimes \\dots \\otimes x_i). $$\nIf $x_1= \\dots=x_i=x$, then we denote this product as $\\lfloor x^i\\rfloor_p $.\nFor $f(\\nu) = \\sum a_i \\nu^i$ a formal series, we call \\textbf{realization of $f(\\nu)$ evaluated at $x$} the sequence\n $$\\left(\\sum_{i=0}^p \\frac{s^i}{i!} \\lfloor x^i \\rfloor_p \\right)_{p \\in {\\mathbb N}}.$$\nFor $f=exp(\\nu)$ in particular, we define\nthe exponential map $\\widehat{exp}(sx)$ to be the sequence in ${\\mathcal U}\\gg[[\\nu]]$\nobtained by taking the realization evaluated at $x$ of the formal series $e^{s\\nu}$.\nBy construction, $\\widehat{exp}(sx)$ is obtained by applying the Schur construction to\n$\\widehat{exp}(s)$ and to the element $x$, and the following theorem\ncan be derived easily from Theorem \\ref{theo:group-like_order_kappa}.\n\n\\begin{thm}\n\\label{theo:group-like_x}\n For all $x \\in {\\gg} $ the exponential sequence $s \\rightarrow \\widehat{exp}(sx)$ is valued in\n the Hom-group $G_{seq}(\\gg) $. Moreover,\n \\begin{enumerate}\n \\item $ \\widehat{exp}(0 x) $ is the unit element $\\mathds{1} \\in G_{seq}(\\gg) $.\n \\item For all $s,t \\in {\\mathbb R}, k \\in {\\mathbb N}$,\n $ \\widehat{exp}(sx) \\vee \\widehat{exp}(tx) = \\alpha(\\widehat{exp}((s+t)x)) = \\widehat{exp}((s+t)\\alpha(x))$.\n \\item $ S(\\widehat{exp}(sx)) = \\widehat{exp}(-sx)$ is a strict inverse, i.e.\n $$ \\widehat{exp}(sx) \\vee \\widehat{exp}(-sx) = \\widehat{exp}(-sx) \\vee \\widehat{exp}(sx) = \\mathds{1} . $$\n \\end{enumerate}\n\\end{thm}\n\nFor a better understanding of $\\widehat{exp}(sx)$, we give its first terms:\n$$\n\\begin{array}{rcl}\n\\widehat{exp}_0(sx)&=& {\\mathds{ 1}} \\\\ \\widehat{exp}_1(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} \\\\\n \\widehat{exp}_2(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$\\alpha(x)$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$x$};\n\\draw (-1,2.2) node {\\scriptsize$x$};\n\\end{tikzpicture} \\\\\n\\widehat{exp}_3(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$\\alpha^2(x)$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$\\alpha(x)$};\n\\draw (-1,2.2) node {\\scriptsize$\\alpha(x)$};\n\\end{tikzpicture}\n+\\frac{s^3 \\nu^3}{3!}\n \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$x$};\n\\draw (0,2.1) node {\\scriptsize$ x$};\n\\draw (-1,2.1) node {\\scriptsize$ \\alpha(x)$};\n\\end{tikzpicture}.\n\\end{array}\n$$\n\nAn immediate consequence of the expression of $ \\widehat{exp}_1(sx) $ is the next proposition, that we invite the reader to see as saying that $G_{seq}(\\gg)$ is large enough to be meaningful.\n\\begin{prop}\\label{prop:expinjective}\nFor every $s \\neq 0$, the assignment\n$x \\mapsto \\widehat{exp}(sx)$\nis an injection from $\\gg$ to the Hom-group of formal group-like sequences $G_{seq}(\\gg)$.\n\\end{prop}\n\nIn order to integrate a Hom-Lie algebra into a Hom-group, we compose the following two functors:\n\\begin{enumerate}\n\\item The functor $ {\\mathcal U} $ from the category of Hom-Lie algebras to the category of\nHom-Hopf algebras which consists in assigning to a Hom-Lie algebra $\\gg$\nits universal algebra ${\\mathcal U}\\gg$, as in Remark \\ref{rem:functor}.\n\\item The functor $G_{seq}$ from the category of Hom-Hopf algebras to the category of Hom-groups,\nwhich consists in assigning to a Hom-Hopf algebra $A$ its formal group-like sequences,\nas in Remark \\ref{rem:functorgroup}.\n\\end{enumerate}\nTherefore the composition of these functors is a functor ${\\mathfrak G}$ from the category of Hom-Lie algebras to the category of Hom-groups. Proposition \\ref{prop:expinjective} implies that this functor is not trivial. Notice that it is compatible with the exponential map in the sense that for every morphism of Hom-Lie algebra $\\varphi: \\gg \\to \\gg'$,\nthe following diagram\n $$ \\xymatrix{ \\gg \\ar[r]^{\\varphi} \\ar[d]^{ \\widehat{exp}(s \\cdot) }& \\gg' \\ar[d]^{ \\widehat{exp}(s \\cdot) }\\\\ {\\mathfrak G} (\\gg) \\ar[r]^{{\\mathfrak G}(\\varphi)}& {\\mathfrak G}(\\gg').} $$\nis commutative.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{INTRO}}\n\n\\noindent{}A normal complex surface $X$ with at worst log terminal\nsingularities, i.e., quotient singularities, is called \\textit{log Del Pezzo}\n\\textit{surface} if its anticanonical divisor $-K_{X}$ is a $\\mathbb{Q}%\n$-Cartier ample divisor. The \\textit{index} of such a surface is defined to\nbe the smallest positive integer $\\ell$ for which $\\ell K_{X}$ is a Cartier\ndivisor. Every log Del \\ Pezzo surface is isomorphic to the\n\\textit{anticanonical model} (in the sense of Sakai \\cite{Sakai}) of the\nrational surface obtained by its minimal desingularization. The following\nTheorem is due to Nikulin \\cite{Nikulin2} (for related results cf. \\cite{Alex, Nikulin3}):\n\n\\begin{theorem}\n\\label{NIKTHM}Let $X$ be a log Del Pezzo surface of index $\\ell$ and\n$\\widetilde{X}\\longrightarrow X$ be its minimal desingularization. Then the\nPicard number $\\rho(\\widetilde{X})$ of $\\widetilde{X}$ \\emph{(}i.e., the rank\nof its Picard group\\emph{)} is bounded by%\n\\begin{equation}\n\\rho(\\widetilde{X})1\\right\\} .$ Its subset $\\{ \\left. \\text{orb}(\\sigma_{F})\\right\\vert\n\\,F\\in\\breve{I}_{Q}\\},$ with $\\breve{I}_{Q}$ defined to be $\\breve{I}%\n_{Q}:=\\left\\{ \\left. F\\in I_{Q}\\, \\right\\vert \\,p_{F}=1\\right\\} ,$ is the\nset of the \\textit{Gorenstein singularities} of $X_{Q}.$\n\nThe minimal desingularization of the surface $X_{Q}$ can be described as\nfollows: Equip the minimal generators of $\\Delta_{Q}$ with an order (e.g.,\nanticlockwise), and assume that for every $F\\in\\mathcal{F}(Q)$ the cone\n$\\sigma_{F}$ has $\\mathbf{n}^{(F)},\\mathbf{n}^{\\prime(F)}\\in\\mathbb{Z}^{2}$ as\nminimal generators ($\\sigma_{F}=\\mathbb{R}_{\\geq0}\\, \\mathbf{n}^{(F)}%\n+\\mathbb{R}_{\\geq0}\\, \\mathbf{n}^{\\prime(F)}$), with $\\mathbf{n}^{(F)}$ coming\nfirst w.r.t. this order. Next, for all $F\\in I_{Q},$ consider the\nnegative-regular continued fraction expansion of\n\\begin{equation}\n\\frac{q_{F}}{q_{F}-p_{F}}=\\left[ \\! \\! \\left[ b_{1}^{(F)},b_{2}^{(F)}%\n,\\ldots,b_{s_{F}}^{(F)}\\right] \\! \\! \\right] :=b_{1}^{(F)}%\n-\\cfrac{1}{b_{2}^{(F)}-\\cfrac{1}{\\begin{array} [c]{cc}\\ddots & \\\\ & -\\cfrac{1}{b_{s_{F}}^{(F)}}\\end{array} }}\\ \\ ,\n\\label{EXPPQCF}%\n\\end{equation}\nand define $\\mathbf{u}_{0}^{(F)}:=\\mathbf{n}^{(F)},$ $\\mathbf{u}_{1}%\n^{(F)}:=\\frac{1}{q_{F}}((q_{F}-p_{F})\\mathbf{n}^{(F)}+\\mathbf{n}^{\\prime\n(F)}),$ and lattice points $\\{ \\mathbf{u}_{j}^{(F)}\\left\\vert \\,2\\leq j\\leq\ns_{F}+1\\right. \\}$ by the formulae\n\\[\n\\mathbf{u}_{j+1}^{(F)}:=b_{j}^{(F)}\\mathbf{u}_{j}^{(F)}-\\mathbf{u}_{j-1}%\n^{(F)},\\ \\ \\forall j\\in\\{1,\\ldots,s_{F}\\}.\\\n\\]\nIt is easy to see that $\\mathbf{u}_{s_{F}+1}^{(F)}=\\mathbf{n}^{\\prime(F)},$\nand that the integers $b_{j}^{(F)}$ are $\\geq2,$ for all $j\\in\\{1,\\ldots\n,s_{F}\\}.$ The singularity orb$(\\sigma_{F})\\in U_{F}$ is resolved minimally by\nthe proper birational map induced by the refinement $\\{ \\mathbb{R}_{\\geq0}\\,\n\\mathbf{u}_{j}^{(F)}+\\mathbb{R}_{\\geq0}\\, \\mathbf{u}_{j+1}^{(F)}\\ \\left\\vert\n\\ 0\\leq j\\leq s_{F}\\right. \\}$ of the fan which is composed of the cone\n$\\sigma_{F}$ and its faces. The exceptional divisor is $E^{(F)}:=%\n{\\textstyle\\sum\\nolimits_{j=1}^{s_{F}}}\nE_{j}^{(F)},$ having%\n\\[\nE_{j}^{(F)}:=\\text{ }\\overline{\\text{orb}(\\mathbb{R}_{\\geq0}\\, \\mathbf{u}%\n_{j}^{(F)})}\\ (\\cong\\mathbb{P}_{\\mathbb{C}}^{1}),\\ \\forall j\\in\\{1,\\ldots\n,s_{F}\\},\n\\]\n(i.e., the closures of the $\\mathbb{T}$-orbits of the \\textquotedblleft\nnew\\textquotedblright\\ rays) as its components, with self-intersection number\n$(E_{j}^{(F)})^{2}=-b_{j}^{(F)}$ (see \\cite[Cor. 1.18 and Prop. 1.19, pp.\n23-25]{Oda}).\n\n\\begin{note}\n(i) If $F\\in\\mathcal{F}(Q),$ and ${\\boldsymbol{\\eta}}_{F}\\in(\\mathbb{Z}%\n^{2})^{\\vee}$ is its unique primitive outer normal vector, we define its\n\\textit{local index} to be the positive integer $l_{F}:=\\left\\langle\n{\\boldsymbol{\\eta}}_{F},F\\right\\rangle ,$ where\n\\[\n\\left\\langle \\cdot,\\cdot\\right\\rangle :\\text{Hom}_{\\mathbb{R}}(\\mathbb{R}%\n^{2},\\mathbb{R})\\times\\mathbb{R}^{2}\\longrightarrow\\mathbb{R}%\n\\]\nis the usual inner product. For $F\\in\\mathcal{F}(Q)\\mathbb{r}I_{Q}$ we have\nobviously $l_{F}=1.$ For $F\\in I_{Q},$ let $K(E^{(F)})$ be the \\textit{local\ncanonical divisor }of the minimal resolution of orb$(\\sigma_{F})\\in U_{F}$ (in\nthe sense of \\cite[p. 75]{Dais}). $K(E^{(F)})$ is a $\\mathbb{Q}$-Cartier\ndivisor (a rational linear combination of $E_{j}^{(F)}$'s), and\n\\begin{equation}\nl_{F}=\\text{ min}\\left\\{ \\left. \\xi\\in\\mathbb{N}\\ \\right\\vert \\ \\xi\nK(E^{(F)})\\text{ is a Cartier divisor}\\right\\} =\\tfrac{q_{F}}{\\text{gcd}%\n(q_{F},p_{F}-1)}. \\label{localind}%\n\\end{equation}\n(ii) If $F\\in I_{Q},$ denoting by $\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F}%\n)}$ the maximal ideal of the local ring $\\mathcal{O}_{X_{Q},\\text{orb}%\n(\\sigma_{F})}$ of the singularity orb$(\\sigma_{F}),$ and by\n\\[\nm_{F}:=\\text{dim}_{\\mathbb{C}}((\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F}%\n)})\/(\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F})}^{2}))-1\n\\]\nits \\textit{multiplicity}, it is known (cf. \\cite[Satz 2.11, p. 347]%\n{Brieskorn}) that\n\\begin{equation}\nm_{F}=2+\\sum_{j=1}^{s_{F}}(b_{j}^{(F)}-2). \\label{multformula}%\n\\end{equation}\n\n\\end{note}\n\n\\begin{lemma}\n\\label{MULTIND}For all $F\\in I_{Q}$ we have\n\\[\nm_{F}\\leq2l_{F}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSee \\cite[Lemma 1.1 (iii), p. 235]{Nikulin1}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{KESQUARE}For all $F\\in I_{Q}$ the self-intersection number of\n$K(E^{(F)})$ equals\n\\[\nK(E^{(F)})^{2}=-\\left( \\frac{2-\\left( p_{F}+\\widehat{p}_{F}\\right) }{q_{F}%\n}+(m_{F}-2)\\right) .\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nFollows from \\cite[Corollary 4.6, p. 96]{Dais} and formula\n(\\ref{multformula}).\n\\end{proof}\n\n\\noindent{}The minimal desingularization $\\varphi:\\widetilde{X}_{Q}%\n\\longrightarrow X_{Q}$ of $X_{Q}$ is constructed by means of the smooth\ncompact toric surface $\\widetilde{X}_{Q}$ which is defined by the fan\n\\[\n\\widetilde{\\Delta}_{Q}:=\\left\\{\n\\begin{array}\n[c]{c}%\n\\text{ the cones }\\left\\{ \\left. \\sigma_{F}\\ \\right\\vert \\ F\\in\n\\mathcal{F}(Q)\\mathbb{r}I_{Q}\\right\\} \\text{ and }\\\\\n\\left\\{ \\left. \\mathbb{R}_{\\geq0}\\, \\mathbf{u}_{j}^{(F)}+\\mathbb{R}_{\\geq\n0}\\, \\mathbf{u}_{j+1}^{(F)}\\ \\right\\vert \\ F\\in I_{Q},\\ j\\in\\{0,1,\\ldots\n,s_{F}\\} \\right\\} ,\\\\\n\\text{together with their faces}%\n\\end{array}\n\\right\\}\n\\]\n(refining each of the cones $\\left\\{ \\left. \\sigma_{F}\\ \\right\\vert \\ F\\in\nI_{Q}\\right\\} $ of $\\Delta_{Q}$ as mentioned above). Furthermore, the\ncorresponding \\textit{discrepancy divisor} equals%\n\\begin{equation}\nK_{\\widetilde{X}_{Q}}-\\varphi^{\\star}K_{X_{Q}}=\\sum_{F\\in I_{Q}}K(E^{(F)}).\n\\label{DISCREPANCY}%\n\\end{equation}\n(By $K_{X_{Q}},K_{\\widetilde{X}_{Q}}$ we denote the canonical divisors of\n$X_{Q}$ and $\\widetilde{X}_{Q},$ respectively.)\n\n\\begin{note}\nBy virtue of (\\ref{localind}) and (\\ref{DISCREPANCY}) the index $\\ell$ of\n$X_{Q}$ (as defined in \\S \\ref{INTRO}) equals%\n\\begin{equation}\n\\ell=\\text{ lcm}\\left\\{ \\left. l_{F}\\ \\right\\vert \\ F\\in\\mathcal{F}%\n(Q)\\right\\} . \\label{LCM}%\n\\end{equation}\n(For simplicity, sometimes $\\ell $ is referred as \\textit{index} of $Q.$) In fact, \\ if we denote by\n\\[\nQ^{\\ast}:=\\left\\{ \\left. \\mathbf{y}\\in\\text{Hom}_{\\mathbb{R}}(\\mathbb{R}%\n^{2},\\mathbb{R})\\ \\right\\vert \\ \\left\\langle \\mathbf{y},\\mathbf{x}%\n\\right\\rangle \\,\\leq\\, 1,\\ \\forall\\, \\mathbf{x}\\in Q\\right\\}\n\\]\nthe \\textit{polar} of the polygon $Q,$ the index $\\ell$ is nothing but\nmin$\\left\\{ \\left. k\\in\\mathbb{N}\\; \\right\\vert \\ \\mathcal{V}(kQ^{\\ast\n})\\subset\\mathbb{Z}^{2}\\right\\} ,$ where $kQ^{\\ast}:=$ $\\left\\{ \\left.\nk\\mathbf{y}\\right\\vert \\mathbf{y}\\in Q^{\\ast}\\right\\} .$ In other words,\n$\\ell$ equals the least common multiple of the (smallest) denominators of the\n(rational) coordinates of the vertices of $Q^{\\ast}.$\n\\end{note}\n\n\\section{Proof of main theorem}\n\n\\noindent{}The proof follows from suitable combination of the two upper bounds\ngiven in Lemmas \\ref{LemmaVQ} and \\ref{LemmaMINDES}. (Henceforth we use freely\nthe notation introduced in \\S \\ref{PRELIM}.)\n\n\\begin{lemma}\n\\label{LemmaVQ}Let $X_{Q}$ be a toric log Del Pezzo surface of index $\\ell\n\\geq1.$ Then%\n\\begin{equation}\n\\sharp(\\mathcal{V}(Q))\\leq4\\, \\text{\\emph{max}}\\left\\{ \\left. l_{H}%\n\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2\\leq4\\ell+2. \\label{firstineq}%\n\\end{equation}\nMoreover, $\\sharp(\\mathcal{V}(Q))= \\,4\\, \\text{\\emph{max}}\\left\\{ \\left.\nl_{H}\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2$, if and only if $\\ell=1$,\nand $Q$ is the unique hexagon \\emph{(}up to lattice-equivalence\\emph{)} with\none interior lattice point. This means, in particular, that for indices\n$\\ell\\geq2$ we have%\n\\begin{equation}\n\\sharp(\\mathcal{V}(Q))\\leq4\\ell+1. \\label{nicebound}%\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nObviously, there exists a facet $F\\in\\mathcal{F}(Q)$ such that $\\sum\n_{\\mathbf{v}\\in\\mathcal{V}(Q)}\\mathbf{v}\\in\\sigma_{F}$ (this is a\n\\emph{special facet}, in the sense of \\cite[Sect.~3]{Oebro}). In addition, since\n$Q$ is two-dimensional, we have for all integers $j$:%\n\\[\n\\sharp\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle\n{\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle =j\\right\\} \\leq2.\n\\]\nWriting $\\mathcal{V}(Q)$ as disjoint union $\\mathcal{V}(Q)=\\mathcal{V}_{\\geq\n0}^{\\left( F\\right) }(Q)\\,%\n{\\textstyle\\bigsqcup}\n\\,\\mathcal{V}_{<0}^{\\left( F\\right) }(Q),$ where%\n\\[\n\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q):=\\left\\{ \\left. \\mathbf{v}%\n\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\geq0\\right\\} \\text{ \\ and \\ \\ }\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q):=\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}%\n(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle\n<0\\right\\} ,\n\\]\nwe observe that\n\\[\n\\sharp(\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q))\\leq2\\left( l_{F}%\n+1\\right) ,\n\\]\nbecause $\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle\n\\in\\{0,1,\\ldots,l_{F}\\}$ for all $\\mathbf{v}\\in\\mathcal{V}_{\\geq0}^{\\left(\nF\\right) }(Q)$. On the other hand,%\n\\begin{align*}\n0 & \\leq\\left\\langle {\\boldsymbol{\\eta}}_{F},\\sum\\nolimits_{\\mathbf{v}%\n\\in\\mathcal{V}(Q)}\\mathbf{v}\\right\\rangle =\\sum\\nolimits_{\\mathbf{v}%\n\\in\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta\n}}_{F},\\mathbf{v}\\right\\rangle +\\sum\\nolimits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\\\\n& =\\sum_{j=0}^{l_{F}}\\ \\sum\\limits_{\\left. \\{\\mathbf{v}\\in\\mathcal{V}%\n_{\\geq0}^{\\left( F\\right) }(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}%\n}_{F},\\mathbf{v}\\right\\rangle =j\\}}\\left\\langle {\\boldsymbol{\\eta}}%\n_{F},\\mathbf{v}\\right\\rangle +\\sum\\limits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\\\\n& \\leq\\sum_{j=0}^{l_{F}}2j+\\sum\\limits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle .\n\\end{align*}\nThis implies%\n\\[\na:=-\\sum\\nolimits_{\\mathbf{v}\\in\\mathcal{V}_{<0}^{\\left( F\\right) }%\n(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle \\leq\n2\\binom{l_{F}+1}{2}.\n\\]\nSetting $\\mu:=\\sharp(\\mathcal{V}_{<0}^{\\left( F\\right) }(Q))$ we examine two\ncases: (i) If $\\mu=2\\lambda,$ for a $\\lambda\\in\\mathbb{N},$ then\n\\[\n\\sum_{j=0}^{\\lambda}2j\\leq a\\Longrightarrow2\\binom{\\lambda+1}{2}\\leq\n2\\binom{l_{F}+1}{2}\\Longrightarrow\\lambda\\leq l_{F}\\text{ and }\\mu\\leq2l_{F}.\n\\]\n(ii) If $\\mu=2\\lambda+1,$ for a $\\lambda\\in\\mathbb{Z}_{\\geq0},$ then\n$\\sum_{j=0}^{\\lambda}2j+\\left( \\lambda+1\\right) \\leq a,$ i.e.,%\n\\[\n2\\binom{\\lambda+1}{2}+\\left( \\lambda+1\\right) \\leq2\\binom{l_{F}+1}%\n{2}\\Longrightarrow\\lambda\\leq l_{F}-1\\text{ and }\\mu\\leq2l_{F}-1.\n\\]\nHence,%\n\\[\n\\sharp(\\mathcal{V}(Q))=\\sharp(\\mathcal{V}_{\\geq0}^{\\left( F\\right)\n}(Q))+\\sharp(\\mathcal{V}_{<0}^{\\left( F\\right) }(Q))\\leq2\\left(\nl_{F}+1\\right) +\\mu\n\\]%\n\\begin{equation}\n\\leq2\\left( l_{F}+1\\right) +2l_{F}=4l_{F}+2\\leq4\\,\\text{max}\\left\\{ \\left.\nl_{H}\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2, \\label{Approx}%\n\\end{equation}\nwith the latter upper bound $\\leq4\\ell+2$ (by (\\ref{LCM})), giving the\ninequality (\\ref{firstineq}). Finally, we deal with the case of equality:\nSuppose that $\\sharp(\\mathcal{V}(Q))=4\\ell^{\\prime}+2$, where\n\\[\n\\ell^{\\prime}:=\\max\\left\\{ \\left. l_{H}\\right\\vert \\,H\\in\\mathcal{F}%\n(Q)\\right\\} .\n\\]\nFrom (\\ref{Approx}) we see that $\\mu=2l_{F}$, and $\\lambda=l_{F}=\\ell^{\\prime\n}$. Therefore, by the equalities in (i) we have for the integers\n$j=-\\ell^{\\prime},\\ldots,0,\\ldots,\\ell^{\\prime}$:\n\\begin{equation}\n\\sharp\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle\n{\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle =j\\right\\} =2.\n\\label{gleichheit}%\n\\end{equation}\nIn particular, $0=\\left\\langle {\\boldsymbol{\\eta}}_{F},\\sum\n\\nolimits_{\\mathbf{v}\\in\\mathcal{V}(Q)}\\mathbf{v}\\right\\rangle $, i.e.,\n$\\sum_{v\\in\\mathcal{V}(Q)}\\mathbf{v}=\\mathbf{0}$. Hence, the previous argument\nholds for \\textit{any} facet. Now let $F^{\\prime}$ be another facet of $Q$\nhaving a common vertex, say $\\mathbf{v,}$ with $F.$ If $\\mathcal{V}%\n(F)=\\{\\mathbf{u},\\mathbf{v}\\}$ and $\\mathcal{V}(F^{\\prime})=\\{\\mathbf{v}%\n,\\mathbf{w}\\},$ then applying (\\ref{gleichheit}) for \\textit{both} $F$ and\n$F^{\\prime}$ we get $\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{w}%\n\\right\\rangle =\\ell^{\\prime}-1$ and $\\left\\langle {\\boldsymbol{\\eta}%\n}_{F^{\\prime}},\\mathbf{u}\\right\\rangle =\\ell^{\\prime}-1$. This implies\n$\\ell^{\\prime}=1=\\ell$, since otherwise the primitive vertex $\\mathbf{v}$\nequals $(\\ell^{\\prime}\/(\\ell^{\\prime}-1))(\\mathbf{w}+\\mathbf{u}-\\mathbf{v})$,\na contradiction. Consequently, $Q$ has to be the unique hexagon (up to\nlattice-equivalence) with just one interior lattice point (see \\cite[Proposition 2.1]{Nill}).\n\\end{proof}\n\n\\begin{lemma}\n\\label{LemmaMINDES}If $X_{Q}$ is a toric log Del Pezzo surface of index\n$\\ell\\geq2$ and $\\widetilde{X}_{Q}\\overset{\\varphi}{\\longrightarrow}X_{Q}$ its\nminimal desingularization, then%\n\\begin{equation}\n\\rho(\\widetilde{X}_{Q})<2\\, \\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1)\n-\\frac{1}{\\ell} \\; \\sharp(\\mathcal{V}(Q)) +10. \\label{secineq}%\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nBy Noether's formula and (\\ref{DISCREPANCY}) we deduce\n\\[\n\\rho(\\widetilde{X}_{Q})=10-K_{\\widetilde{X}_{Q}}^{2}=10-K_{X_{Q}}^{2}%\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}.\n\\]\nSince $-\\ell K_{X_{Q}}$ is an ample Cartier divisor on $X_{Q},$ we can compute\nby \\cite[Proposition 2.10, p. 79]{Oda} its self-intersection number:\n\\[\n(-\\ell K_{X_{Q}})^{2}=2\\text{\\thinspace area}(\\ell Q^{\\ast})\\Longrightarrow\nK_{X_{Q}}^{2}=\\frac{2}{\\ell^{2}}\\,\\text{area}(\\ell Q^{\\ast})=2\\text{\\thinspace\narea}(Q^{\\ast}).\n\\]\nFor any facet $H$ of $\\ell Q^{\\ast}$ the primitive outer normal vector is given by\nsome vertex of $Q$, i.e., the lattice distance of $H$ from $\\mathbf{0}$ equals\n$\\ell$. This implies\n\\[\n\\text{area}(\\ell Q^{\\ast}) \\geq\\frac{1}{2} \\ell\\;\\sharp(\\mathcal{F}(\\ell\nQ^{\\ast})) = \\frac{1}{2} \\ell\\;\\sharp(\\mathcal{V}(Q)).\n\\]\nHence,\n\\[\n-K_{X_{Q}}^{2} = -\\frac{2}{\\ell^{2}} \\;\\text{area}(\\ell Q^{\\ast}) \\leq\n-\\frac{1}{\\ell} \\;\\sharp(\\mathcal{V}(Q)).\n\\]\nOn the other hand, by Lemma \\ref{KESQUARE} we infer that%\n\\[\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}=\\sum_{F\\in I_{Q}}\\left( \\frac{2-\\left(\np_{F}+\\widehat{p}_{F}\\right) }{q_{F}}+(m_{F}-2)\\right) .\n\\]\nTaking into account that $m_{F}=2$ for all $F\\in\\breve{I}_{Q},$ and that\n$p_{F}+\\widehat{p}_{F}\\geq2$ for all $F\\in I_{Q},$ which is valid as equality\nonly for $p_{F}=\\widehat{p}_{F}=1,$ i.e., whenever $F\\in\\breve{I}_{Q},$ we\nobtain%\n\\[\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}=-\\sum_{F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}%\n}K(E^{(F)})^{2}<\\sum_{F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}}(m_{F}-2)\n\\]%\n\\begin{align*}\n& \\leq\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\,\\,\\text{max}\\left\\{ \\left.\nm_{F}-2 \\;\\right\\vert \\; F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}\\right\\} \\\\\n& \\leq\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\,\\,\\text{max}\\left\\{ \\left.\n2(l_{F}-1) \\;\\right\\vert \\; F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}\\right\\}\n\\leq2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1),\n\\end{align*}\nwhere the last but one inequality follows from Lemma \\ref{MULTIND}. Thus,\n$\\rho(\\widetilde{X}_{Q})$ is strictly smaller than the sum $10 -\\sharp\n(\\mathcal{V}(Q))\/\\ell+ 2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1).$\n\\end{proof}\n\n\\noindent{}\\textit{Proof of Theorem \\ref{main}}. If $\\ell=1,$ then\n$\\rho(\\widetilde{X}_{Q})\\leq7$ by the known classification of the reflexive\npolygons (see \\cite{KS} or \\cite[Proposition 2.1]{Nill}). If $\\ell\\geq2,$\napplying (\\ref{nicebound}) and (\\ref{secineq}), and the inequality\n$\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\leq\\sharp(\\mathcal{V}(Q)),$ we get\n\\[\n\\rho(\\widetilde{X}_{Q})<2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell\n-1)-\\frac{1}{\\ell}\\;\\sharp(\\mathcal{V}(Q))+10\n\\]%\n\\[\n\\leq\\sharp(\\mathcal{V}(Q))\\left( 2(\\ell-1)-\\frac{1}{\\ell}\\right)\n+10\\leq\\left( 4\\ell+1\\right) \\left( 2(\\ell-1)-\\frac{1}{\\ell}\\right) +10,\n\\]\ni.e., $\\rho(\\widetilde{X}_{Q})<8\\ell^{2}-6\\ell+4-\\frac{1}{\\ell}$, which yields the bound for $\\ell \\geq 2$.\n\\hfill{}$\\square$\n\n\\section{Discussion, improvements and examples}\n\n\\noindent{} First, let us note that from the proof of Theorem \\ref{main} we derive a \\emph{linear} upper bound on\n$\\rho(\\widetilde{X}_{Q})$, if the number of vertices of $Q$ is \\emph{fixed}. It is\ntherefore natural to ask for an example of an infinite family $\\{Q_{i}\\}$ of\nLDP-polygons with increasing number of vertices, for which $\\rho(\\widetilde\n{X}_{Q_{i}})$ exhibits a non-linear growth with respect to the indices of its members.\nTo the best knowledge of the authors, this seems to be an open question.\n\nNow, in some specific cases we can further improve the bound (\\ref{mainbound}).\nIf $Q$ is an LDP-polygon and $F\\in I_{Q},$ then, according to (\\ref{localind}%\n), there is a positive integer $\\beta_{F}$ such that%\n\\[\np_{F}-1=\\beta_{F}\\cdot\\frac{q_{F}}{l_{F}}\\Longrightarrow l_{F}\\,(p_{F}%\n-1)=\\beta_{F}\\,q_{F}.\n\\]\nSince $l_{F}(p_{F}-1) \\nu}\\sum_{\\sigma,\\sigma'}U'_{\\mu\\nu}n_{i\\mu\\sigma} n_{i\\nu\\sigma'}\n\\right.\\nonumber\\\\\n&&\\left.-\\sum_{\\mu\\neq\\nu}J_{\\mu\\nu}\\Vec{S}_{i\\mu}\\cdot\\Vec{S}_{i\\nu}\n+\\sum_{\\mu\\neq\\nu}J'_{\\mu\\nu}\nc_{i\\mu\\uparrow}^\\dagger c_{i\\mu\\downarrow}^\\dagger\nc_{i\\nu\\downarrow}c_{i\\nu\\uparrow}\n\\right).\n\\end{eqnarray}\nWe apply the fluctuation exchange (FLEX) approximation\\cite{Bickers1989,Dahm} \nusing multiorbital \nHubbard Hamiltonian. In FLEX, bubble and ladder type diagrams consisting of \nrenormalized Green's functions are summed up to obtain the susceptibilities, \nwhich are used to calculate the self energy. The renormalized Green's \nfunctions are then determined self-consistently from the Dyson's equation.\nThe obtained Green's function is plugged into the linearized Eliashberg \nequation, whose eigenvalue $\\lambda$ reaches unity at the superconducting \ntransition temperature $T=T_c$. Also, in order to investigate the correlation \nbetween superconductivity and magnetism, we obtain the Stoner factor $a_S$\nof the antiferromagnetism at the wave vector $(\\pi,0)$ in the unfolded \nBrillouin zone, which is defined as the largest eigenvalue of the matrix \n$U\\chi_0({\\bf{k}}=(\\pi,0),i\\omega_n=0)$, where $U$ is the interaction and \n$\\chi_0$ is the irreducible susceptibility matrices, respectively. \nThis value monitors the tendency towards stripe type antiferromagnetism and \nthe strength of the spin fluctuations at zero energy. Since the three \ndimensionality is not strong in Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$, \nwe take a two dimensional model where \nwe neglect the out-of-plane hopping integrals, and take $32 \\times 32$ \n$k$-point meshes and 4096 Matsubara frequencies.\n\nAs for the electron-electron interaction values, \nwe adopt the orbital-dependent interactions\nas obtained from first principles calculation \nin ref.\\onlinecite{Miyake_private} \nfor Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$, but multiply all of them \nby a constant reducing \nfactor $f$. The reason for introducing this factor is as follows.\nAs has been studied in refs.\\onlinecite{Ikeda_prb,Arita,Ikeda_jpsj} \nthe FLEX calculation for models obtained from LDA calculations tends to \noverestimate the effect of the \nself-energy because LDA already partially takes into account the \neffect of the self-energy in the exchange-correlation functional. \nWhen the electron-electron interactions as large as those evaluated \nfrom first principles are adopted in the FLEX calculation, \nthis double counting of the self-energy becomes so large \nthat the band structure largely differs from its original one. \nIn such a case, the spin fluctuations will develop around the wave vector\n$(\\pi,\\pi)$ rather than $(\\pi,0)$, which is in disagreement \nwith the experimental observations. \nIn the present study, we therefore \nintroduce the factor $f$ so as to reduce the electron-electron interactions,\nwhile maintaining the relative magnitude between interactions of different \norbitals.\n\n\\subsection{Bond angle}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig6}\n\\caption{(a) The Eliashberg equation eigenvalue for \nsuperconductivity ($s\\pm$-wave pairing) (solid) and the Stoner factor \nat $(\\pi,0)$ (dashed) against the bond angle for \ntemperature $T = 0.005$. The interaction \nreduction factor is $f = 0.45$.\\label{fig:6}}\n\\end{figure}\n\nWe show the eigenvalue of the Eliashberg equation $\\lambda$ for \nthe s$\\pm$-wave superconductivity and the Stoner factor at $(\\pi,0)$ for \nthe hypothetical lattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ varying \nthe bond angle while fixing the bond length(Fig.\\ref{fig:6}).\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig7}\n\\caption{The arrows indicate the wave vector of the \ndominant pairing interactions for the (a)$X^2-Y^2$ and \n(b) $XZ\/YZ$ portions of the Fermi surface in the case where \nthe inner hole Fermi surface ($\\alpha_1$) is barely present. \nIn this case, $\\alpha_1$ is a mixture of $X^2-Y^2$ and $XZ\/YZ$.\\label{fig:7}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig8}\n\\caption{The gap function obtained by FLEX for the hypothetical \nlattice structures of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$. The bond angle \n$\\alpha$ is set to 110$^\\circ$ or 111$^\\circ$, while the bond length \nis fixed at the original value.\\label{fig:8}}\n\\end{figure}\n\nAs we decrease the bond angle from 115 to 110$^\\circ$, \neigenvalue of the Eliashberg equation $\\lambda$ increases, reflecting \nthe appearance of the $\\gamma$ Fermi surface around $(\\pi,\\pi)$.\nSuperconductivity is locally optimized around 110$^\\circ$, \nbut $\\lambda$ immediately goes down for larger bond angle. \nThis is in contrast to the case of LaFeAsO, \nwhere $\\lambda$ is broadly maximized around the regular tetrahedron \nbond angle.\nThis difference can be understood from the comparison between \nFig.\\ref{fig:2}(c) and Fig.\\ref{fig:4}(b).\nNamely, in the case of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ with hypothetical \nbond angle, the Fermi surface configuration (b) \nwith the optimal Fermi surface configuration \nis missing, i.e., in the three Fermi surface regime, \n$\\alpha_1$ Fermi surface around $(0,0)$ is \nconstructed from a mixture of $X^2-Y^2$ and $XZ\/YZ$ orbital \ncharacters. In this configuration, The pair scattering takes place not only \nat $\\sim (\\pi,0)$ but also at $\\sim (\\pi,\\pi)$ \ndue to the same orbital character between \n$\\alpha_2$ and $\\gamma$ Fermi surfaces.\nSince these Fermi surfaces interact with repulsive pairing interactions, \na frustration arises in the sign of the superconducting gap \nas shown schematically in Fig.\\ref{fig:7}. \nIn addition to this, there can also be some $XZ\/YZ$ component remaining in the \n$\\alpha_1$ Fermi surface, and this portion tends to change the sign from\nthe $\\beta$ Fermi surfaces, making it another possible factor\nfor the frustration.\nThe effect of the frustration appears in the form of the superconducting gap.\nIn Fig.\\ref{fig:8}, we show the gap function for \nthe hypothetical lattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nat the bond angles 110$^\\circ$ and 111$^\\circ$. \nThe sign of the gap function on $\\alpha_1$ is positive at 111$^\\circ$, \nbut is very small (barely positive) at 110$^\\circ$\\cite{commentSUST},\nreflecting the effect of the frustration.\nThe bond angle of 110$^\\circ$ is actually very close to that of \nCa$_4$Al$_2$O$_6$Fe$_2$P$_2$, so the appearance of a very small gap at \nthis bond angle may be related to the nodal gap structure \nsuggested experimentally for Ca$_4$Al$_2$O$_6$Fe$_2$P$_2$\\cite{Kinouchi2011}.\nAs the bond angle is further reduced, the $\\alpha_1$ Fermi surface \ndisappears but the effect of the frustration remains strong as far as the \ntop of the $\\alpha_1$ hole band does not sink far below the \nFermi level. In fact, the frustration effect can be very strong \nright after the Fermi surface disappears because the top of \nthis $\\alpha_1$ band (the closest point to the Fermi level) \nhas pure $X^2-Y^2$ orbital character.\nTherefore, $\\lambda$ is suppressed around the bond angle of \n105$^\\circ\\sim$ 108$^\\circ$. Meanwhile, the Fermi surface nesting itself\nbecomes very good in this regime because there are now two hole and two \nelectron Fermi surfaces with no doped carriers, so that the \naverage area of the hole and the electron Fermi surfaces becomes the same.\nIn particular, around the bond angle of 105$^\\circ$, the nesting \nbecomes nearly perfect, as shown in Fig.\\ref{fig:9}. Therefore, \nthe Stoner factor at $(\\pi,0)$ takes a local maximum around this bond angle.\nAs the bond angle is reduced even further, the $X^2-Y^2$ band \nsinks far below the Fermi level and the frustration effect \nbecomes small, so that $\\lambda$ increases once again to a value \ncomparable to that around the local maximum around the regular \ntetrahedron bond angle.\nAt the same time, the Fermi surface nesting becomes somewhat degraded, \nand the Stoner factor is reduced. \nFor smaller bond angle$<96^\\circ$ (which may not be realistic), \nthe Fermi surface becomes too large, and the \nsuperconductivity is degraded. The bottom line here is that \nsuperconductivity is favored at around two bond angles 102$^\\circ$ and \n110$^\\circ$, and antiferromagnetism is favored in the regime in between \nthese angles.\nThis is at least qualitatively consistent with the experimental \nobservations for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$.\n\nThe important point here is that \nsuperconductivity is suppressed in the intermediate bond angle regime \ndue to the frustration effect. Apart from this, \nantiferromagnetism is favored around this bond angle \nregime due to a nearly perfect nesting of the Fermi surface.\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig9}\n\\caption{The Fermi surface of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nfor the hypothetical lattice structures with $\\alpha=105^\\circ$ \nand $102^\\circ$ (solid), superposed with the Fermi surface \nshifted by $(\\pi,0)$ (dashed).\\label{fig:9}}\n\\end{figure}\n\n\n\\subsection{Pnictogen height}\n\n\\begin{figure}\n\\includegraphics[width=6.5cm]{fig10}\n\\caption{The pnictogen height dependence of (a) the Eliashberg equation \neigenvalue and (b) the Stoner factor at $(\\pi,0)$ for the hypothetical \nlattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$. \nSeveral values of the reducing factor are taken for comparison.\\label{fig:10}\n(c) A schematic figure of the $x$ dependence of $\\lambda$ for \nsuperconductivity and $a_S$ for antiferromagnetism.}\n\\end{figure}\n\nWe have studied in the previous section the bond angle dependence of \nsuperconductivity and the spin fluctuations, and mentioned the \npossible relation between the calculation results and \nthe experimental observations for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$.\nAs mentioned previously, \nthe actual lattice structure variation upon replacing As by P is \nmore close to the variance of the \npnictogen height $h_{\\rm Pn}$ rather than just the bond angle. \nThe increase of the bond length results in an increase in the density of \nstates, generally resulting in an enhancement of \nboth superconductivity and spin fluctuations\\cite{Usui2011}.\nIn Fig.\\ref{fig:10}, \nwe show the eigenvalue of the Eliashberg equation and \nthe Stoner factor at $(\\pi,0)$ for the hypothetical lattice \nstructure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nvarying solely the pnictogen height $h_{\\rm Pn}$. \nAround $h_{\\rm Pn}=1.3\\sim 1.35{\\rm \\AA}$, corresponding to the \nP content close to unity, \nthe height dependence of $\\lambda$ is weak (or $\\lambda$ is even suppressed \nwith the increase of $h_{\\rm Pn}$ for large $f$),\nwhile the Stoner factor rapidly increases with $h_{\\rm Pn}$. \nThis height regime corresponds to the bond angle regime of \n$110^\\circ\\sim 108^\\circ$, where \nsuperconductivity is suppressed due to the momentum space \nfrustration, and at the same time antiferromagnetism \nis favored due to the nearly perfect nesting (Fig.\\ref{fig:7}). \nHere in Fig.\\ref{fig:10}(a), \nthe enhancement of superconductivity \nby the increase of the density of states is canceled out due to the \nfrustration effect, so that the $h_{\\rm Pn}$ dependence of $\\lambda$ is weak.\nOn the other hand, the Stoner factor quickly grows due to the \ncooperation of the \ngood nesting and the increased density of states.\nAs the pnictogen height increases further beyond $1.35{\\rm \\AA}$, \n$\\lambda$ starts to \nincrease rapidly due to the reduction of the frustration \nand the increase of the \ndensity of states, while the Stoner factor tends to saturate because \nthe nearly perfect nesting is degraded.\nThis overall tendency is summarized in a schematic figure in \nFig.\\ref{fig:10}(c)\n\n\n\\section{Pressure experiment}\nOur theoretical study so far has \nshown that in the region where antiferromagnetism \nappears in the phase diagram, not only antiferromagnetism is \nenhanced due to the good Fermi surface nesting, but also superconductivity is \nsuppressed due to the momentum space frustration, and these two are \nindependent matters. Since superconductivity is suppressed \nregardless of whether antiferromagnetism is present or not,\nsuperconductivity may not take place \neven when antiferromagnetism is suppressed by applying pressure, as \nis often done in other iron based superconductors. \n\nTo actually see this experimentally, \nwe have applied hydrostatic pressure to \nCa$_4$Al$_2$O$_6$Fe$_2$(As$_{1-x}$P$_x$)$_2$. \nThe results are shown in Fig.\\ref{fig:11}.\nFor the end compounds $x=0$ and $x=1$, $T_c$ \nmonotonically decreases with increasing pressure.\nThis is most likely due to the decrease in the density of states.\nFor $x=0.75$, where antiferromagnetism takes place at ambient pressure, \nsuperconductivity is not found up to 12GPa, although the \nantiferromagnetic transition is smeared out at high pressures. \nThis is in contrast with cases where antiferromagnetism takes place at \nambient pressure, but gives way to superconductivity under \npressure.\nThe present experimental result supports the scenario that \nsuperconductivity in the intermediate $x$ regime is suppressed \nby momentum space frustration, apart from the presence of the \nantiferromagnetism itself.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig11}\n\\caption{(a) The pressure dependence of the superconducting \ntransition temperature for various materials. The resistivity \nagainst pressure for Ca$_4$Al$_2$O$_6$Fe$_2$(As$_{1-x}$P$_x$)$_2$ for \n(b) $x=0$, (c)$x=0.75$ and (d)$x=1$. \\label{fig:11}}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn the present paper, we studied the origin of the peculiar phase \ndiagram obtained for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$ \nusing a five orbital model constructed from first principles \nband calculation.\nWhile the inner hole Fermi surface is absent at $x=0$\\cite{Miyake2010}, \nit is present at $x=1$, but the orbital character has strong\n $X^2-Y^2$ character rather than $XZ\/YZ$ as in LaFeAsO. \nThis gives rise to momentum space frustration of the \npairing interaction mediated by spin fluctuations, and degrades \nsuperconductivity. We propose this to be one of the reasons why \n$T_c$ is not so high in Ca$_4$Al$_2$O$_6$Fe$_2$P despite of the \nmaximized multiplicity of the hole Fermi surface.\nThe frustration effect remains strong \neven after the inner Fermi surface has disappeared for $x<1$ \nbecause the top of the band with $X^2-Y^2$ orbital character \nremains near the Fermi level. At the same time, the \ndisappearance of the most inner hole Fermi surface gives \nvery good nesting of the electron and hole Fermi surfaces due to the \nequal number of sheets, favoring antiferromagnetism in the \nintermediate regime of $x$. Finally for $x\\sim 1$, the top of the \nband sinks far below the Fermi level, and the frustration effect \nis reduced, so that superconductivity is favored once again.\nAlthough we cannot directly determine which one of the \nsuperconductivity and antiferromagnetism wins, the tendency \nobserved in the calculation is at least consistent with the \nexperimental observation, where nodeless and nodal superconducting \nphases are separated by an antiferromagnetic phase.\nFinally, we have performed hydrostatic pressure experiment, \nwhich further supports our scenario that superconductivity \nis suppressed by momentum space frustration in the intermediate \n$x$ regime.\n\n\\section{ACKNOWLEDGMENTS}\n\nWe are grateful to H. Mukuda, H. Kinouchi, and Y.Kitaoka \nfor fruitful discussions.\nThe numerical calculations were performed at the Supercomputer Center, \nISSP, University of Tokyo. This study has been supported by \nGrants-in-Aid for Scientific Research from JSPS. \nK.S acknowledges support from JSPS.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}