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)[source]#

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 filtered modules 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

>>> from sage.all import *
>>> C = LieAlgebras(ZZ).WithBasis().FiniteDimensional().Graded(); C
Category of finite dimensional graded Lie algebras with basis over Integer Ring
>>> C.super_categories()
[Category of graded Lie algebras with basis over Integer Ring,
 Category of finite dimensional filtered modules with basis over Integer Ring,
 Category of finite dimensional Lie algebras with basis over Integer Ring]

>>> C is LieAlgebras(ZZ).WithBasis().FiniteDimensional().Graded()
True
class ParentMethods[source]#

Bases: object

homogeneous_component_as_submodule(d)[source]#

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}},                         # needs sage.combinat sage.modules
....:                     nilpotent=True, category=C)
sage: L.homogeneous_component_as_submodule(2)                           # needs sage.combinat sage.modules
Sparse vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[0 0 1]

>>> from sage.all import *
>>> C = LieAlgebras(QQ).WithBasis().Graded()
>>> C = C.FiniteDimensional().Stratified().Nilpotent()
>>> L = LieAlgebra(QQ, {('x','y'): {'z': Integer(1)}},                         # needs sage.combinat sage.modules
...                     nilpotent=True, category=C)
>>> L.homogeneous_component_as_submodule(Integer(2))                           # needs 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)[source]#

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

>>> from sage.all import *
>>> C = LieAlgebras(QQ).WithBasis().Graded().Stratified().FiniteDimensional()
>>> 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

>>> from sage.all import *
>>> C is C.Nilpotent()
True
class ParentMethods[source]#

Bases: object

degree_on_basis(m)[source]#

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: # needs sage.combinat sage.modules
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

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> C = LieAlgebras(QQ).WithBasis().Graded()
>>> C = C.FiniteDimensional().Stratified().Nilpotent()
>>> sc = {('X','Y'): {'Z': Integer(1)}}
>>> L = LieAlgebra(QQ, sc, nilpotent=True, category=C, names=('X', 'Y', 'Z',)); (X, Y, Z,) = L._first_ngens(3)
>>> L.degree_on_basis(X.leading_support())
1
>>> X.degree()
1
>>> Y.degree()
1
>>> L[X, Y]
Z
>>> Z.degree()
2