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 - is_abelian()[source]¶
- Return - Truesince- selfis 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 - Truesince- selfis 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