import itertools
from .cartan_type import Standard_Cartan
from sympy.core.backend import eye, Rational
from sympy.core.singleton import S
[docs]
class TypeE(Standard_Cartan):
def __new__(cls, n):
if n < 6 or n > 8:
raise ValueError("Invalid value of n")
return Standard_Cartan.__new__(cls, "E", 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("E6")
>>> c.dimension()
8
"""
return 8
[docs]
def basic_root(self, i, j):
"""
This is a method just to generate roots
with a -1 in the ith position and a 1
in the jth position.
"""
root = [0]*8
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.
This method returns the ith simple root for E_n.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("E6")
>>> c.simple_root(2)
[1, 1, 0, 0, 0, 0, 0, 0]
"""
n = self.n
if i == 1:
root = [-0.5]*8
root[0] = 0.5
root[7] = 0.5
return root
elif i == 2:
root = [0]*8
root[1] = 1
root[0] = 1
return root
else:
if i in (7, 8) and n == 6:
raise ValueError("E6 only has six simple roots!")
if i == 8 and n == 7:
raise ValueError("E7 only has seven simple roots!")
return self.basic_root(i - 3, i - 2)
[docs]
def positive_roots(self):
"""
This method generates all the positive roots of
A_n. This is half of all of the roots of E_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
neghalf = Rational(-1, 2)
poshalf = S.Half
if n == 6:
posroots = {}
k = 0
for i in range(n-1):
for j in range(i+1, n-1):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
root = [poshalf, poshalf, poshalf, poshalf, poshalf,
neghalf, neghalf, poshalf]
for a, b, c, d, e in itertools.product(
range(2), range(2), range(2), range(2), range(2)):
if (a + b + c + d + e)%2 == 0:
k += 1
if a == 1:
root[0] = neghalf
if b == 1:
root[1] = neghalf
if c == 1:
root[2] = neghalf
if d == 1:
root[3] = neghalf
if e == 1:
root[4] = neghalf
posroots[k] = root[:]
return posroots
if n == 7:
posroots = {}
k = 0
for i in range(n-1):
for j in range(i+1, n-1):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
k += 1
posroots[k] = [0, 0, 0, 0, 0, 1, 1, 0]
root = [poshalf, poshalf, poshalf, poshalf, poshalf,
neghalf, neghalf, poshalf]
for a, b, c, d, e, f in itertools.product(
range(2), range(2), range(2), range(2), range(2), range(2)):
if (a + b + c + d + e + f)%2 == 0:
k += 1
if a == 1:
root[0] = neghalf
if b == 1:
root[1] = neghalf
if c == 1:
root[2] = neghalf
if d == 1:
root[3] = neghalf
if e == 1:
root[4] = neghalf
if f == 1:
root[5] = poshalf
posroots[k] = root[:]
return posroots
if n == 8:
posroots = {}
k = 0
for i in range(n):
for j in range(i+1, n):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
root = [poshalf, poshalf, poshalf, poshalf, poshalf,
neghalf, neghalf, poshalf]
for a, b, c, d, e, f, g in itertools.product(
range(2), range(2), range(2), range(2), range(2),
range(2), range(2)):
if (a + b + c + d + e + f + g)%2 == 0:
k += 1
if a == 1:
root[0] = neghalf
if b == 1:
root[1] = neghalf
if c == 1:
root[2] = neghalf
if d == 1:
root[3] = neghalf
if e == 1:
root[4] = neghalf
if f == 1:
root[5] = poshalf
if g == 1:
root[6] = poshalf
posroots[k] = root[:]
return posroots
[docs]
def roots(self):
"""
Returns the total number of roots of E_n
"""
n = self.n
if n == 6:
return 72
if n == 7:
return 126
if n == 8:
return 240
[docs]
def cartan_matrix(self):
"""
Returns 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('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)
for i in range(3, n - 1):
m[i, i+1] = -1
m[i, i-1] = -1
m[0, 2] = m[2, 0] = -1
m[1, 3] = m[3, 1] = -1
m[2, 3] = -1
m[n-1, n-2] = -1
return m
[docs]
def basis(self):
"""
Returns the number of independent generators of E_n
"""
n = self.n
if n == 6:
return 78
if n == 7:
return 133
if n == 8:
return 248
def dynkin_diagram(self):
n = self.n
diag = " "*8 + str(2) + "\n"
diag += " "*8 + "0\n"
diag += " "*8 + "|\n"
diag += " "*8 + "|\n"
diag += "---".join("0" for i in range(1, n)) + "\n"
diag += "1 " + " ".join(str(i) for i in range(3, n+1))
return diag