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

sage: LieAlgebras(QQ).Graded().Stratified().FiniteDimensional()
Category of finite dimensional stratified Lie algebras over Rational Field

Python

>>> from sage.all import *
>>> LieAlgebras(QQ).Graded().Stratified().FiniteDimensional()
Category of finite dimensional stratified Lie algebras over Rational Field

extra_super_categories()[source]#

Implements the fact that a finite dimensional stratified Lie algebra is nilpotent.

EXAMPLES:

Sage

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

Python

>>> 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

sage: LieAlgebras(QQ).Graded().Stratified()
Category of stratified Lie algebras over Rational Field

Python

>>> from sage.all import *
>>> LieAlgebras(QQ).Graded().Stratified()
Category of stratified Lie algebras over Rational Field