from sympy.core import Rational
from sympy.core.compatibility import range
from .cartan_type import Standard_Cartan
from sympy.matrices import Matrix
[docs]class TypeF(Standard_Cartan):
def __new__(cls, n):
if n != 4:
raise ValueError("n should be 4")
return Standard_Cartan.__new__(cls, "F", 4)
[docs] def dimension(self):
"""Dimension of the vector space V underlying the Lie algebra
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("F4")
>>> c.dimension()
4
"""
return 4
[docs] def basic_root(self, i, j):
"""Generate roots with 1 in ith position and -1 in jth postion
"""
n = self.n
root = [0]*n
root[i] = 1
root[j] = -1
return root
[docs] def simple_root(self, i):
"""The ith simple root of F_4
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("F4")
>>> c.simple_root(3)
[0, 0, 0, 1]
"""
if i < 3:
return basic_root(i-1, i)
if i == 3:
root = [0]*4
root[3] = 1
return root
if i == 4:
root = [Rational(-1, 2)]*4
return root
[docs] def positive_roots(self):
"""Generate all the positive roots of A_n
This is half of all of the roots of F_4; 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-1):
for j in range(i+1, n):
k += 1
posroots[k] = self.basic_root(i, j)
k += 1
root = self.basic_root(i, j)
root[j] = 1
posroots[k] = root
for i in range(0, n):
k += 1
root = [0]*n
root[i] = 1
posroots[k] = root
k += 1
root = [Rational(1, 2)]*n
posroots[k] = root
for i in range(1, 4):
k += 1
root = [Rational(1, 2)]*n
root[i] = Rational(-1, 2)
posroots[k] = root
posroots[k+1] = [Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2)]
posroots[k+2] = [Rational(1, 2), Rational(-1, 2), Rational(1, 2), Rational(-1, 2)]
posroots[k+3] = [Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]
posroots[k+4] = [Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(-1, 2)]
return posroots
[docs] def roots(self):
"""
Returns the total number of roots for F_4
"""
return 48
[docs] def cartan_matrix(self):
"""The Cartan matrix for F_4
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('A4')
>>> c.cartan_matrix()
Matrix([
[ 2, -1, 0, 0],
[-1, 2, -1, 0],
[ 0, -1, 2, -1],
[ 0, 0, -1, 2]])
"""
m = Matrix( 4, 4, [2, -1, 0, 0, -1, 2, -2, 0, 0,
-1, 2, -1, 0, 0, -1, 2])
return m
[docs] def basis(self):
"""
Returns the number of independent generators of F_4
"""
return 52
def dynkin_diagram(self):
diag = "0---0=>=0---0\n"
diag += " ".join(str(i) for i in range(1, 5))
return diag
