Abelian Lie Algebras¶

AUTHORS:

• Travis Scrimshaw (2016-06-07): Initial version
class sage.algebras.lie_algebras.abelian.AbelianLieAlgebra(R, names, index_set, category, **kwds)

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
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)

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
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