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)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

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

Bases: object

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)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

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

Bases: object

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)
sage: L.degree_on_basis(X.leading_support())
1
sage: X.degree()
1
sage: Y.degree()
1
sage: L[X, Y]
Z
sage: Z.degree()
2