Source code for sympy.liealgebras.type_a

from __future__ import print_function, division

from sympy.core.compatibility import range
from sympy.liealgebras.cartan_type import Standard_Cartan
from sympy.core.backend import eye

[docs]class TypeA(Standard_Cartan): """ This class contains the information about the A series of simple Lie algebras. ==== """ def __new__(cls, n): if n < 1: raise ValueError("n can not be less than 1") return Standard_Cartan.__new__(cls, "A", n)
[docs] def dimension(self): """Dimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A4") >>> c.dimension() 5 """ return self.n+1
[docs] def basic_root(self, i, j): """ This is a method just to generate roots with a 1 iin the ith position and a -1 in the jth position. """ n = self.n root = *(n+1) root[i] = 1 root[j] = -1 return root
[docs] def simple_root(self, i): """ Every lie algebra has a unique root system. Given a root system Q, there is a subset of the roots such that an element of Q is called a simple root if it cannot be written as the sum of two elements in Q. If we let D denote the set of simple roots, then it is clear that every element of Q can be written as a linear combination of elements of D with all coefficients non-negative. In A_n the ith simple root is the root which has a 1 in the ith position, a -1 in the (i+1)th position, and zeroes elsewhere. This method returns the ith simple root for the A series. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A4") >>> c.simple_root(1) [1, -1, 0, 0, 0] """ return self.basic_root(i-1, i)
[docs] def positive_roots(self): """ This method generates all the positive roots of A_n. This is half of all of the roots of A_n; by multiplying all the positive roots by -1 we get the negative roots. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A3") >>> c.positive_roots() {1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0], 5: [0, 1, 0, -1], 6: [0, 0, 1, -1]} """ n = self.n posroots = {} k = 0 for i in range(0, n): for j in range(i+1, n+1): k += 1 posroots[k] = self.basic_root(i, j) return posroots
[docs] def highest_root(self): """ Returns the highest weight root for A_n """ return self.basic_root(0, self.n)
[docs] def roots(self): """ Returns the total number of roots for A_n """ n = self.n return n*(n+1)
[docs] def cartan_matrix(self): """ Returns the Cartan matrix for A_n. The Cartan matrix matrix for a Lie algebra is generated by assigning an ordering to the simple roots, (alpha, ...., alpha[l]). Then the ijth entry of the Cartan matrix is (<alpha[i],alpha[j]>). Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType('A4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -1], [ 0, 0, -1, 2]]) """ n = self.n m = 2 * eye(n) i = 1 while i < n-1: m[i, i+1] = -1 m[i, i-1] = -1 i += 1 m[0,1] = -1 m[n-1, n-2] = -1 return m
[docs] def basis(self): """ Returns the number of independent generators of A_n """ n = self.n return n**2 - 1
[docs] def lie_algebra(self): """ Returns the Lie algebra associated with A_n """ n = self.n return "su(" + str(n + 1) + ")"
def dynkin_diagram(self): n = self.n diag = "---".join("0" for i in range(1, n+1)) + "\n" diag += " ".join(str(i) for i in range(1, n+1)) return diag