Graded Lie Algebras¶
AUTHORS:
- Eero Hakavuori (2018-08-16): initial version 
- class sage.categories.graded_lie_algebras.GradedLieAlgebras(base_category)[source]¶
- Bases: - GradedModulesCategory- Category of graded Lie algebras. - class Stratified(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- Category of stratified Lie algebras. - A graded Lie algebra \(L = \bigoplus_{k=1}^M L_k\) (where possibly \(M = \infty\)) is called stratified if it is generated by \(L_1\); in other words, we have \(L_{k+1} = [L_1, L_k]\). - class FiniteDimensional(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- Category of finite dimensional stratified Lie algebras. - EXAMPLES: - sage: LieAlgebras(QQ).Graded().Stratified().FiniteDimensional() Category of finite dimensional stratified Lie algebras over Rational Field - >>> from sage.all import * >>> LieAlgebras(QQ).Graded().Stratified().FiniteDimensional() Category of finite dimensional stratified Lie algebras over Rational Field - extra_super_categories()[source]¶
- Implement the fact that a finite dimensional stratified Lie algebra is nilpotent. - EXAMPLES: - sage: C = LieAlgebras(QQ).Graded().Stratified().FiniteDimensional() sage: C.extra_super_categories() [Category of nilpotent Lie algebras over Rational Field] sage: C is C.Nilpotent() True sage: C.is_subcategory(LieAlgebras(QQ).Nilpotent()) True - >>> from sage.all import * >>> C = LieAlgebras(QQ).Graded().Stratified().FiniteDimensional() >>> C.extra_super_categories() [Category of nilpotent Lie algebras over Rational Field] >>> C is C.Nilpotent() True >>> C.is_subcategory(LieAlgebras(QQ).Nilpotent()) True 
 
 
 - class SubcategoryMethods[source]¶
- Bases: - object- Stratified()[source]¶
- Return the full subcategory of stratified objects of - self.- A Lie algebra is stratified if it is graded and generated as a Lie algebra by its component of degree one. - EXAMPLES: - sage: LieAlgebras(QQ).Graded().Stratified() Category of stratified Lie algebras over Rational Field - >>> from sage.all import * >>> LieAlgebras(QQ).Graded().Stratified() Category of stratified Lie algebras over Rational Field