# -*- coding: utf-8 -*-
from .cartan_type import Standard_Cartan
from sympy.matrices import Matrix
[docs]class TypeG(Standard_Cartan):
def __new__(cls, n):
if n != 2:
raise ValueError("n should be 2")
return Standard_Cartan.__new__(cls, "G", 2)
[docs] def dimension(self):
"""Dimension of the vector space V underlying the Lie algebra
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("G2")
>>> c.dimension()
3
"""
return 3
[docs] def simple_root(self, i):
"""The ith simple root of G_2
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.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("G2")
>>> c.simple_root(1)
[0, 1, -1]
"""
if i == 1:
return [0, 1, -1]
else:
return [1, -2, 1]
[docs] def positive_roots(self):
"""Generate 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]}
"""
roots = {1: [0, 1, -1], 2: [1, -2, 1], 3: [1, -1, 0], 4: [1, 0, 1],
5: [1, 1, -2], 6: [2, -1, -1]}
return roots
[docs] def roots(self):
"""
Returns the total number of roots of G_2"
"""
return 12
[docs] def cartan_matrix(self):
"""The Cartan matrix for G_2
The Cartan matrix matrix for a Lie algebra is
generated by assigning an ordering to the simple
roots, (alpha[1], ...., 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("G2")
>>> c.cartan_matrix()
Matrix([
[ 2, -1],
[-3, 2]])
"""
m = Matrix( 2, 2, [2, -1, -3, 2])
return m
[docs] def basis(self):
"""
Returns the number of independent generators of G_2
"""
return 14
def dynkin_diagram(self):
diag = "0≡<≡0\n1 2"
return diag
