Abelian Lie Algebras

AUTHORS:

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

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

Bases: sage.algebras.lie_algebras.structure_coefficients.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

class Element

Bases: sage.algebras.lie_algebras.structure_coefficients.LieAlgebraWithStructureCoefficients.Element

is_abelian()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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

is_nilpotent()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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

is_solvable()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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

class sage.algebras.lie_algebras.abelian.InfiniteDimensionalAbelianLieAlgebra(R, index_set, prefix='L', **kwds)

Bases: sage.algebras.lie_algebras.lie_algebra.InfinitelyGeneratedLieAlgebra, sage.structure.indexed_generators.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

Bases: sage.algebras.lie_algebras.lie_algebra_element.LieAlgebraElement

dimension()

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

EXAMPLES:

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

is_abelian()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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

is_nilpotent()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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

is_solvable()

Return True since self is an abelian Lie algebra.

EXAMPLES:

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