from sympy.core import Rational
from sympy.core.compatibility import range
from .cartan_type import Standard_Cartan
from sympy.matrices import eye
[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 postion.
"""
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 == 7 or i == 8 and n == 6:
raise ValueError("E6 only has six simple roots!")
if i == 8 and n == 7:
raise ValueError("E7 has only 7 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
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 = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2),
Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
if (a + b + c + d + e)%2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
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 = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2),
Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
for f in range(0, 2):
if (a + b + c + d + e + f)%2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
if f == 1:
root[5] = Rational(1, 2)
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 = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2),
Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
for f in range(0, 2):
for g in range(0, 2):
if (a + b + c + d + e + f + g)%2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
if f == 1:
root[5] = Rational(1, 2)
if g == 1:
root[6] = Rational(1, 2)
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)
i = 3
while i < n-1:
m[i, i+1] = -1
m[i, i-1] = -1
i += 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
