def so(R, n, representation='bracket'):
r"""
The Lie algebra `\mathfrak{so}_n`.
The Lie algebra `\mathfrak{so}_n` is the type `B_k` Lie algebra
if `n = 2k - 1` or the type `D_k` Lie algebra if `n = 2k`, and in
either case is finite dimensional. As a matrix Lie algebra, it
is given by the set of all real anti-symmetric `n \times n` matrices.
INPUT:
- ``R`` -- the base ring
- ``n`` -- the size of the matrix
- ``representation`` -- (default: ``'bracket'``) can be one of
the following:
* ``'bracket'`` - use brackets and the Chevalley basis
* ``'matrix'`` - use matrices
EXAMPLES:
We first construct `\mathfrak{so}_5` using the Chevalley basis::
sage: so5 = lie_algebras.so(QQ, 5); so5
Lie algebra of ['B', 2] in the Chevalley basis
sage: E1,E2, F1,F2, H1,H2 = so5.gens()
sage: so5([E1, [E1, E2]])
0
sage: X = so5([E2, [E2, E1]]); X
-2*E[alpha[1] + 2*alpha[2]]
sage: H1.bracket(X)
0
sage: H2.bracket(X)
-4*E[alpha[1] + 2*alpha[2]]
sage: so5([H1, [E1, E2]])
-E[alpha[1] + alpha[2]]
sage: so5([H2, [E1, E2]])
0
We do the same construction of `\mathfrak{so}_4` using the Chevalley
basis::
sage: so4 = lie_algebras.so(QQ, 4); so4
Lie algebra of ['D', 2] in the Chevalley basis
sage: E1,E2, F1,F2, H1,H2 = so4.gens()
sage: H1.bracket(E1)
2*E[alpha[1]]
sage: H2.bracket(E1) == so4.zero()
True
sage: E1.bracket(E2) == so4.zero()
True
We now construct `\mathfrak{so}_4` as a matrix Lie algebra::
sage: sl2 = lie_algebras.sl(QQ, 2, representation='matrix')
sage: E1,E2, F1,F2, H1,H2 = so4.gens()
sage: H2.bracket(E1) == so4.zero()
True
sage: E1.bracket(E2) == so4.zero()
True
"""
if representation == 'bracket':
from sage.algebras.lie_algebras.classical_lie_algebra import LieAlgebraChevalleyBasis
if n % 2 == 0:
return LieAlgebraChevalleyBasis(R, ['D', n//2])
else:
return LieAlgebraChevalleyBasis(R, ['B', (n-1)//2])
if representation == 'matrix':
from sage.algebras.lie_algebras.classical_lie_algebra import so as so_matrix
return so_matrix(R, n)
raise ValueError("invalid representation")
def so(R, n, representation='bracket'):
r"""
The Lie algebra `\mathfrak{so}_n`.
The Lie algebra `\mathfrak{so}_n` is the type `B_k` Lie algebra
if `n = 2k - 1` or the type `D_k` Lie algebra if `n = 2k`, and in
either case is finite dimensional. As a matrix Lie algebra, it
is given by the set of all real anti-symmetric `n \times n` matrices.
INPUT:
- ``R`` -- the base ring
- ``n`` -- the size of the matrix
- ``representation`` -- (default: ``'bracket'``) can be one of
the following:
* ``'bracket'`` - use brackets and the Chevalley basis
* ``'matrix'`` - use matrices
EXAMPLES:
We first construct `\mathfrak{so}_5` using the Chevalley basis::
sage: so5 = lie_algebras.so(QQ, 5); so5
Lie algebra of ['B', 2] in the Chevalley basis
sage: E1,E2, F1,F2, H1,H2 = so5.gens()
sage: so5([E1, [E1, E2]])
0
sage: X = so5([E2, [E2, E1]]); X
-2*E[alpha[1] + 2*alpha[2]]
sage: H1.bracket(X)
0
sage: H2.bracket(X)
-4*E[alpha[1] + 2*alpha[2]]
sage: so5([H1, [E1, E2]])
-E[alpha[1] + alpha[2]]
sage: so5([H2, [E1, E2]])
0
We do the same construction of `\mathfrak{so}_4` using the Chevalley
basis::
sage: so4 = lie_algebras.so(QQ, 4); so4
Lie algebra of ['D', 2] in the Chevalley basis
sage: E1,E2, F1,F2, H1,H2 = so4.gens()
sage: H1.bracket(E1)
2*E[alpha[1]]
sage: H2.bracket(E1) == so4.zero()
True
sage: E1.bracket(E2) == so4.zero()
True
We now construct `\mathfrak{so}_4` as a matrix Lie algebra::
sage: sl2 = lie_algebras.sl(QQ, 2, representation='matrix')
sage: E1,E2, F1,F2, H1,H2 = so4.gens()
sage: H2.bracket(E1) == so4.zero()
True
sage: E1.bracket(E2) == so4.zero()
True
"""
if representation == 'bracket':
from sage.algebras.lie_algebras.classical_lie_algebra import LieAlgebraChevalleyBasis
if n % 2 == 0:
return LieAlgebraChevalleyBasis(R, ['D', n//2])
else:
return LieAlgebraChevalleyBasis(R, ['B', (n-1)//2])
if representation == 'matrix':
from sage.algebras.lie_algebras.classical_lie_algebra import so as so_matrix
return so_matrix(R, n)
raise ValueError("invalid representation")