# 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]#

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]#

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]#
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