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:
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 ofself
as a submodule.EXAMPLES:
sage: C = LieAlgebras(QQ).WithBasis().Graded() sage: C = C.FiniteDimensional().Stratified().Nilpotent() sage: L = LieAlgebra(QQ, {('x','y'): {'z': 1}}, # optional - sage.combinat sage.modules ....: nilpotent=True, category=C) sage: L.homogeneous_component_as_submodule(2) # optional - sage.combinat sage.modules Sparse vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 0 1]
- class Stratified(base_category)#
Bases:
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) # optional - sage.combinat sage.modules sage: L.degree_on_basis(X.leading_support()) # optional - sage.combinat sage.modules 1 sage: X.degree() # optional - sage.combinat sage.modules 1 sage: Y.degree() # optional - sage.combinat sage.modules 1 sage: L[X, Y] # optional - sage.combinat sage.modules Z sage: Z.degree() # optional - sage.combinat sage.modules 2