Source code for sympy.liealgebras.type_d

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

[docs]class TypeD(Standard_Cartan):

def __new__(cls, n):
if n < 3:
raise ValueError("n cannot be less than 3")
return Standard_Cartan.__new__(cls, "D", n)

[docs]    def dimension(self):
"""Dmension of the vector space V underlying the Lie algebra

Examples
========

>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("D4")
>>> c.dimension()
4
"""

return self.n

[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 = [0]*n
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 D_n, the first n-1 simple roots are the same as
the roots in A_(n-1) (a 1 in the ith position, a -1
in the (i+1)th position, and zeroes elsewhere).
The nth simple root is the root in which there 1s in
the nth and (n-1)th positions, and zeroes elsewhere.

This method returns the ith simple root for the D series.

Examples
========

>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("D4")
>>> c.simple_root(2)
[0, 1, -1, 0]

"""

n = self.n
if i < n:
return self.basic_root(i-1, i)
else:
root = [0]*n
root[n-2] = 1
root[n-1] = 1
return root

[docs]    def positive_roots(self):
"""
This method generates all the positive roots of
A_n.  This is half of all of the roots of D_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-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
return posroots

[docs]    def roots(self):
"""
Returns the total number of roots for D_n"
"""

n = self.n
return 2*n*(n-1)

[docs]    def cartan_matrix(self):
"""
Returns the Cartan matrix for D_n.
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('D4')
>>> c.cartan_matrix()
Matrix([
[ 2, -1,  0,  0],
[-1,  2, -1, -1],
[ 0, -1,  2,  0],
[ 0, -1,  0,  2]])

"""

n = self.n
m = 2*eye(n)
i = 1
while i < n-2:
m[i,i+1] = -1
m[i,i-1] = -1
i += 1
m[n-2, n-3] = -1
m[n-3, n-1] = -1
m[n-1, n-3] = -1
m[0, 1] = -1
return m

[docs]    def basis(self):
"""
Returns the number of independent generators of D_n
"""
n = self.n
return n*(n-1)/2

[docs]    def lie_algebra(self):
"""
Returns the Lie algebra associated with D_n"
"""

n = self.n
return "so(" + str(2*n) + ")"

def dynkin_diagram(self):
n = self.n
diag = " "*4*(n-3) + str(n-1) + "\n"
diag += " "*4*(n-3) + "0\n"
diag += " "*4*(n-3) +"|\n"
diag += " "*4*(n-3) + "|\n"
diag += "---".join("0" for i in range(1,n)) + "\n"
diag += "   ".join(str(i) for i in range(1, n-1)) + "   "+str(n)
return diag