from .cartan_type import Standard_Cartan
from sympy.core.compatibility import range
from sympy.matrices import eye
[docs]class TypeC(Standard_Cartan):
def __new__(cls, n):
if n < 3:
raise ValueError("n can not be less than 3")
return Standard_Cartan.__new__(cls, "C", 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("C3")
>>> c.dimension()
3
"""
n = self.n
return n
[docs] def basic_root(self, i, j):
"""Generate roots with 1 in ith position and a -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 for the C series
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 C_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 is a 2 in
the nth position and zeroes elsewhere.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("C3")
>>> c.simple_root(2)
[0, 1, -1]
"""
n = self.n
if i < n:
return self.basic_root(i-1,i)
else:
root = [0]*self.n
root[n-1] = 2
return root
[docs] def positive_roots(self):
"""Generates all the positive roots of A_n
This is half of all of the roots of C_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
for i in range(0, n):
k += 1
root = [0]*n
root[i] = 2
posroots[k] = root
return posroots
[docs] def roots(self):
"""
Returns the total number of roots for C_n"
"""
n = self.n
return 2*(n**2)
[docs] def cartan_matrix(self):
"""The Cartan matrix for C_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('C4')
>>> c.cartan_matrix()
Matrix([
[ 2, -1, 0, 0],
[-1, 2, -1, 0],
[ 0, -1, 2, -1],
[ 0, 0, -2, 2]])
"""
n = self.n
m = 2 * eye(n)
i = 1
while i < n-1:
m[i, i+1] = -1
m[i, i-1] = -1
i += 1
m[0,1] = -1
m[n-1, n-2] = -2
return m
[docs] def basis(self):
"""
Returns the number of independent generators of C_n
"""
n = self.n
return n*(2*n + 1)
[docs] def lie_algebra(self):
"""
Returns the Lie algebra associated with C_n"
"""
n = self.n
return "sp(" + str(2*n) + ")"
def dynkin_diagram(self):
n = self.n
diag = "---".join("0" for i in range(1, n)) + "=<=0\n"
diag += " ".join(str(i) for i in range(1, n+1))
return diag
