Abelian Lie Algebras

AUTHORS:

  • Travis Scrimshaw (2016-06-07): Initial version

class sage.algebras.lie_algebras.abelian.AbelianLieAlgebra(R, names, index_set, category, **kwds)[source]

Bases: LieAlgebraWithStructureCoefficients

An abelian Lie algebra.

A Lie algebra \(\mathfrak{g}\) is abelian if \([x, y] = 0\) for all \(x, y \in \mathfrak{g}\).

EXAMPLES:

sage: L.<x, y> = LieAlgebra(QQ, abelian=True)
sage: L.bracket(x, y)
0

>>> from sage.all import *
>>> L = LieAlgebra(QQ, abelian=True, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.bracket(x, y)
0
class Element[source]

Bases: Element

is_abelian()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = LieAlgebra(QQ, 3, 'x', abelian=True)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(3), 'x', abelian=True)
>>> L.is_abelian()
True
is_nilpotent()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = LieAlgebra(QQ, 3, 'x', abelian=True)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(3), 'x', abelian=True)
>>> L.is_abelian()
True
is_solvable()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = LieAlgebra(QQ, 3, 'x', abelian=True)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(3), 'x', abelian=True)
>>> L.is_abelian()
True
class sage.algebras.lie_algebras.abelian.InfiniteDimensionalAbelianLieAlgebra(R, index_set, prefix='L', **kwds)[source]

Bases: InfinitelyGeneratedLieAlgebra, IndexedGenerators

An infinite dimensional abelian Lie algebra.

A Lie algebra \(\mathfrak{g}\) is abelian if \([x, y] = 0\) for all \(x, y \in \mathfrak{g}\).

class Element[source]

Bases: LieAlgebraElement

dimension()[source]

Return the dimension of self, which is \(\infty\).

EXAMPLES:

sage: L = lie_algebras.abelian(QQ, index_set=ZZ)
sage: L.dimension()
+Infinity

>>> from sage.all import *
>>> L = lie_algebras.abelian(QQ, index_set=ZZ)
>>> L.dimension()
+Infinity
is_abelian()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = lie_algebras.abelian(QQ, index_set=ZZ)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = lie_algebras.abelian(QQ, index_set=ZZ)
>>> L.is_abelian()
True
is_nilpotent()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = lie_algebras.abelian(QQ, index_set=ZZ)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = lie_algebras.abelian(QQ, index_set=ZZ)
>>> L.is_abelian()
True
is_solvable()[source]

Return True since self is an abelian Lie algebra.

EXAMPLES:

sage: L = lie_algebras.abelian(QQ, index_set=ZZ)
sage: L.is_abelian()
True

>>> from sage.all import *
>>> L = lie_algebras.abelian(QQ, index_set=ZZ)
>>> L.is_abelian()
True