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| from .cartan_type import CartanType | |
| from sympy.core.basic import Atom | |
| class RootSystem(Atom): | |
| """Represent the root system of a simple Lie algebra | |
| Every simple Lie algebra has a unique root system. To find the root | |
| system, we first consider the Cartan subalgebra of g, which is the maximal | |
| abelian subalgebra, and consider the adjoint action of g on this | |
| subalgebra. There is a root system associated with this action. Now, a | |
| root system over a vector space V is a set of finite vectors Phi (called | |
| roots), which satisfy: | |
| 1. The roots span V | |
| 2. The only scalar multiples of x in Phi are x and -x | |
| 3. For every x in Phi, the set Phi is closed under reflection | |
| through the hyperplane perpendicular to x. | |
| 4. If x and y are roots in Phi, then the projection of y onto | |
| the line through x is a half-integral multiple of x. | |
| Now, there is a subset of Phi, which we will call Delta, such that: | |
| 1. Delta is a basis of V | |
| 2. Each root x in Phi can be written x = sum k_y y for y in Delta | |
| The elements of Delta are called the simple roots. | |
| Therefore, we see that the simple roots span the root space of a given | |
| simple Lie algebra. | |
| References | |
| ========== | |
| .. [1] https://en.wikipedia.org/wiki/Root_system | |
| .. [2] Lie Algebras and Representation Theory - Humphreys | |
| """ | |
| def __new__(cls, cartantype): | |
| """Create a new RootSystem object | |
| This method assigns an attribute called cartan_type to each instance of | |
| a RootSystem object. When an instance of RootSystem is called, it | |
| needs an argument, which should be an instance of a simple Lie algebra. | |
| We then take the CartanType of this argument and set it as the | |
| cartan_type attribute of the RootSystem instance. | |
| """ | |
| obj = Atom.__new__(cls) | |
| obj.cartan_type = CartanType(cartantype) | |
| return obj | |
| def simple_roots(self): | |
| """Generate the simple roots of the Lie algebra | |
| The rank of the Lie algebra determines the number of simple roots that | |
| it has. This method obtains the rank of the Lie algebra, and then uses | |
| the simple_root method from the Lie algebra classes to generate all the | |
| simple roots. | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> roots = c.simple_roots() | |
| >>> roots | |
| {1: [1, -1, 0, 0], 2: [0, 1, -1, 0], 3: [0, 0, 1, -1]} | |
| """ | |
| n = self.cartan_type.rank() | |
| roots = {} | |
| for i in range(1, n+1): | |
| root = self.cartan_type.simple_root(i) | |
| roots[i] = root | |
| return roots | |
| def all_roots(self): | |
| """Generate all the roots of a given root system | |
| The result is a dictionary where the keys are integer numbers. It | |
| generates the roots by getting the dictionary of all positive roots | |
| from the bases classes, and then taking each root, and multiplying it | |
| by -1 and adding it to the dictionary. In this way all the negative | |
| roots are generated. | |
| """ | |
| alpha = self.cartan_type.positive_roots() | |
| keys = list(alpha.keys()) | |
| k = max(keys) | |
| for val in keys: | |
| k += 1 | |
| root = alpha[val] | |
| newroot = [-x for x in root] | |
| alpha[k] = newroot | |
| return alpha | |
| def root_space(self): | |
| """Return the span of the simple roots | |
| The root space is the vector space spanned by the simple roots, i.e. it | |
| is a vector space with a distinguished basis, the simple roots. This | |
| method returns a string that represents the root space as the span of | |
| the simple roots, alpha[1],...., alpha[n]. | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> c.root_space() | |
| 'alpha[1] + alpha[2] + alpha[3]' | |
| """ | |
| n = self.cartan_type.rank() | |
| rs = " + ".join("alpha["+str(i) +"]" for i in range(1, n+1)) | |
| return rs | |
| def add_simple_roots(self, root1, root2): | |
| """Add two simple roots together | |
| The function takes as input two integers, root1 and root2. It then | |
| uses these integers as keys in the dictionary of simple roots, and gets | |
| the corresponding simple roots, and then adds them together. | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> newroot = c.add_simple_roots(1, 2) | |
| >>> newroot | |
| [1, 0, -1, 0] | |
| """ | |
| alpha = self.simple_roots() | |
| if root1 > len(alpha) or root2 > len(alpha): | |
| raise ValueError("You've used a root that doesn't exist!") | |
| a1 = alpha[root1] | |
| a2 = alpha[root2] | |
| newroot = [_a1 + _a2 for _a1, _a2 in zip(a1, a2)] | |
| return newroot | |
| def add_as_roots(self, root1, root2): | |
| """Add two roots together if and only if their sum is also a root | |
| It takes as input two vectors which should be roots. It then computes | |
| their sum and checks if it is in the list of all possible roots. If it | |
| is, it returns the sum. Otherwise it returns a string saying that the | |
| sum is not a root. | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> c.add_as_roots([1, 0, -1, 0], [0, 0, 1, -1]) | |
| [1, 0, 0, -1] | |
| >>> c.add_as_roots([1, -1, 0, 0], [0, 0, -1, 1]) | |
| 'The sum of these two roots is not a root' | |
| """ | |
| alpha = self.all_roots() | |
| newroot = [r1 + r2 for r1, r2 in zip(root1, root2)] | |
| if newroot in alpha.values(): | |
| return newroot | |
| else: | |
| return "The sum of these two roots is not a root" | |
| def cartan_matrix(self): | |
| """Cartan matrix of Lie algebra associated with this root system | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> c.cartan_matrix() | |
| Matrix([ | |
| [ 2, -1, 0], | |
| [-1, 2, -1], | |
| [ 0, -1, 2]]) | |
| """ | |
| return self.cartan_type.cartan_matrix() | |
| def dynkin_diagram(self): | |
| """Dynkin diagram of the Lie algebra associated with this root system | |
| Examples | |
| ======== | |
| >>> from sympy.liealgebras.root_system import RootSystem | |
| >>> c = RootSystem("A3") | |
| >>> print(c.dynkin_diagram()) | |
| 0---0---0 | |
| 1 2 3 | |
| """ | |
| return self.cartan_type.dynkin_diagram() | |