Lambda Bracket Algebras With Basis

AUTHORS:

• Reimundo Heluani (2020-08-21): Initial implementation.

class sage.categories.lambda_bracket_algebras_with_basis.LambdaBracketAlgebrasWithBasis(base_category)

Bases: CategoryWithAxiom_over_base_ring

The category of Lambda bracket algebras with basis.

EXAMPLES:

Sage

sage: LieConformalAlgebras(QQbar).WithBasis()                                   # needs sage.rings.number_field
Category of Lie conformal algebras with basis over Algebraic Field

Python

>>> from sage.all import *
>>> LieConformalAlgebras(QQbar).WithBasis()                                   # needs sage.rings.number_field
Category of Lie conformal algebras with basis over Algebraic Field

class ElementMethods

Bases: object

index()

The index of this basis element.

EXAMPLES:

Sage

sage: # needs sage.combinat sage.modules
sage: V = lie_conformal_algebras.NeveuSchwarz(QQ)
sage: V.inject_variables()
Defining L, G, C
sage: G.T(3).index()
('G', 3)
sage: v = V.an_element(); v
L + G + C
sage: v.index()
Traceback (most recent call last):
...
ValueError: index can only be computed for monomials, got L + G + C

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> V = lie_conformal_algebras.NeveuSchwarz(QQ)
>>> V.inject_variables()
Defining L, G, C
>>> G.T(Integer(3)).index()
('G', 3)
>>> v = V.an_element(); v
L + G + C
>>> v.index()
Traceback (most recent call last):
...
ValueError: index can only be computed for monomials, got L + G + C

class FinitelyGeneratedAsLambdaBracketAlgebra(base_category)

Bases: CategoryWithAxiom_over_base_ring

The category of finitely generated lambda bracket algebras with basis.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: C = LieConformalAlgebras(QQbar)
sage: C1 = C.WithBasis().FinitelyGenerated(); C1
Category of finitely generated Lie conformal algebras with basis
 over Algebraic Field
sage: C2 = C.FinitelyGenerated().WithBasis(); C2
Category of finitely generated Lie conformal algebras with basis
 over Algebraic Field
sage: C1 is C2
True

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> C = LieConformalAlgebras(QQbar)
>>> C1 = C.WithBasis().FinitelyGenerated(); C1
Category of finitely generated Lie conformal algebras with basis
 over Algebraic Field
>>> C2 = C.FinitelyGenerated().WithBasis(); C2
Category of finitely generated Lie conformal algebras with basis
 over Algebraic Field
>>> C1 is C2
True

class Graded(base_category)

Bases: GradedModulesCategory

The category of H-graded finitely generated lambda bracket algebras with basis.

EXAMPLES:

Sage

sage: C = LieConformalAlgebras(QQbar)                                   # needs sage.rings.number_field
sage: C.WithBasis().FinitelyGenerated().Graded()                        # needs sage.rings.number_field
Category of H-graded finitely generated Lie conformal algebras
 with basis over Algebraic Field

Python

>>> from sage.all import *
>>> C = LieConformalAlgebras(QQbar)                                   # needs sage.rings.number_field
>>> C.WithBasis().FinitelyGenerated().Graded()                        # needs sage.rings.number_field
Category of H-graded finitely generated Lie conformal algebras
 with basis over Algebraic Field

class ParentMethods

Bases: object

degree_on_basis(m)

Return the degree of the basis element indexed by m in self.

EXAMPLES:

Sage

sage: V = lie_conformal_algebras.Virasoro(QQ)                   # needs sage.combinat sage.modules
sage: V.degree_on_basis(('L', 2))                               # needs sage.combinat sage.modules
4

Python

>>> from sage.all import *
>>> V = lie_conformal_algebras.Virasoro(QQ)                   # needs sage.combinat sage.modules
>>> V.degree_on_basis(('L', Integer(2)))                               # needs sage.combinat sage.modules
4