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:
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
- is_abelian()#
Return
True
sinceself
is an abelian Lie algebra.EXAMPLES:
sage: L = LieAlgebra(QQ, 3, 'x', abelian=True) sage: L.is_abelian() True
- is_nilpotent()#
Return
True
sinceself
is an abelian Lie algebra.EXAMPLES:
sage: L = LieAlgebra(QQ, 3, 'x', abelian=True) sage: L.is_abelian() True
- is_solvable()#
Return
True
sinceself
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:
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#
Bases:
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
sinceself
is an abelian Lie algebra.EXAMPLES:
sage: L = lie_algebras.abelian(QQ, index_set=ZZ) sage: L.is_abelian() True
- is_nilpotent()#
Return
True
sinceself
is an abelian Lie algebra.EXAMPLES:
sage: L = lie_algebras.abelian(QQ, index_set=ZZ) sage: L.is_abelian() True
- is_solvable()#
Return
True
sinceself
is an abelian Lie algebra.EXAMPLES:
sage: L = lie_algebras.abelian(QQ, index_set=ZZ) sage: L.is_abelian() True