# Finite Dimensional Graded Lie Algebras With Basis¶

AUTHORS:

• Eero Hakavuori (2018-08-16): initial version
class sage.categories.finite_dimensional_graded_lie_algebras_with_basis.FiniteDimensionalGradedLieAlgebrasWithBasis(base_category)

Category of finite dimensional graded Lie algebras with a basis.

A grading of a Lie algebra $$\mathfrak{g}$$ is a direct sum decomposition $$\mathfrak{g} = \bigoplus_{i} V_i$$ such that $$[V_i,V_j] \subset V_{i+j}$$.

EXAMPLES:

sage: C = LieAlgebras(ZZ).WithBasis().FiniteDimensional().Graded(); C
Category of finite dimensional graded lie algebras with basis over Integer Ring
sage: C.super_categories()
[Category of graded lie algebras with basis over Integer Ring,
Category of finite dimensional lie algebras with basis over Integer Ring]

sage: C is LieAlgebras(ZZ).WithBasis().FiniteDimensional().Graded()
True

class ParentMethods
homogeneous_component_as_submodule(d)

Return the d-th homogeneous component of self as a submodule.

EXAMPLES:

sage: C = LieAlgebras(QQ).WithBasis().Graded()
sage: C = C.FiniteDimensional().Stratified().Nilpotent()
sage: L = LieAlgebra(QQ, {('x','y'): {'z': 1}},
....:                     nilpotent=True, category=C)
sage: L.homogeneous_component_as_submodule(2)
Sparse vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[0 0 1]

class Stratified(base_category)

Category of finite dimensional stratified Lie algebras with a basis.

A stratified Lie algebra is a graded Lie algebra that is generated as a Lie algebra by its homogeneous component of degree 1. That is to say, for a graded Lie algebra $$L = \bigoplus_{k=1}^M L_k$$, we have $$L_{k+1} = [L_1, L_k]$$.

EXAMPLES:

sage: C = LieAlgebras(QQ).WithBasis().Graded().Stratified().FiniteDimensional()
sage: C
Category of finite dimensional stratified lie algebras with basis over Rational Field


A finite-dimensional stratified Lie algebra is nilpotent:

sage: C is C.Nilpotent()
True

class ParentMethods
degree_on_basis(m)

Return the degree of the basis element indexed by m.

If the degrees of the basis elements are not defined, they will be computed. By assumption the stratification $$L_1 \oplus \cdots \oplus L_s$$ of self is such that each component $$L_k$$ is spanned by some subset of the basis.

The degree of a basis element $$X$$ is therefore the largest index $$k$$ such that $$X \in L_k \oplus \cdots \oplus L_s$$. The space $$L_k \oplus \cdots \oplus L_s$$ is by assumption the $$k$$-th term of the lower central series.

EXAMPLES:

sage: C = LieAlgebras(QQ).WithBasis().Graded()
sage: C = C.FiniteDimensional().Stratified().Nilpotent()
sage: sc = {('X','Y'): {'Z': 1}}
sage: L.<X,Y,Z> = LieAlgebra(QQ, sc, nilpotent=True, category=C)