Finitely Generated Lambda bracket Algebras

AUTHORS:

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

class sage.categories.finitely_generated_lambda_bracket_algebras.FinitelyGeneratedLambdaBracketAlgebras(base_category)

Bases: CategoryWithAxiom_over_base_ring

The category of finitely generated lambda bracket algebras.

EXAMPLES:

Sage

sage: from sage.categories.lambda_bracket_algebras import LambdaBracketAlgebras
sage: LambdaBracketAlgebras(QQbar).FinitelyGenerated()                          # needs sage.rings.number_field
Category of finitely generated lambda bracket algebras over Algebraic Field

Python

>>> from sage.all import *
>>> from sage.categories.lambda_bracket_algebras import LambdaBracketAlgebras
>>> LambdaBracketAlgebras(QQbar).FinitelyGenerated()                          # needs sage.rings.number_field
Category of finitely generated lambda bracket algebras over Algebraic Field

class Graded(base_category)

Bases: GradedModulesCategory

The category of H-graded finitely generated Lie conformal algebras.

EXAMPLES:

Sage

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

Python

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

class ParentMethods

Bases: object

gen(i)

The i-th generator of this Lie conformal algebra.

EXAMPLES:

Sage

sage: # needs sage.combinat sage.modules
sage: V = lie_conformal_algebras.Affine(QQ, 'A1')
sage: V.gens()
(B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K'])
sage: V.gen(0)
B[alpha[1]]
sage: V.1
B[alphacheck[1]]

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> V = lie_conformal_algebras.Affine(QQ, 'A1')
>>> V.gens()
(B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K'])
>>> V.gen(Integer(0))
B[alpha[1]]
>>> V.gen(1)
B[alphacheck[1]]

ngens()

The number of generators of this Lie conformal algebra.

EXAMPLES:

Sage

sage: Vir = lie_conformal_algebras.Virasoro(QQ)                         # needs sage.combinat sage.modules
sage: Vir.ngens()                                                       # needs sage.combinat sage.modules
2

sage: V = lie_conformal_algebras.Affine(QQ, 'A2')                       # needs sage.combinat sage.modules
sage: V.ngens()                                                         # needs sage.combinat sage.modules
9

Python

>>> from sage.all import *
>>> Vir = lie_conformal_algebras.Virasoro(QQ)                         # needs sage.combinat sage.modules
>>> Vir.ngens()                                                       # needs sage.combinat sage.modules
2

>>> V = lie_conformal_algebras.Affine(QQ, 'A2')                       # needs sage.combinat sage.modules
>>> V.ngens()                                                         # needs sage.combinat sage.modules
9

some_elements()

Some elements of this Lie conformal algebra.

This method returns a list with elements containing at least the generators.

EXAMPLES:

Sage

sage: V = lie_conformal_algebras.Affine(QQ, 'A1',                       # needs sage.combinat sage.modules
....:                                   names=('e', 'h', 'f'))
sage: V.some_elements()                                                 # needs sage.combinat sage.modules
[e, h, f, K, ...]
sage: all(v.parent() is V for v in V.some_elements())                   # needs sage.combinat sage.modules
True

Python

>>> from sage.all import *
>>> V = lie_conformal_algebras.Affine(QQ, 'A1',                       # needs sage.combinat sage.modules
...                                   names=('e', 'h', 'f'))
>>> V.some_elements()                                                 # needs sage.combinat sage.modules
[e, h, f, K, ...]
>>> all(v.parent() is V for v in V.some_elements())                   # needs sage.combinat sage.modules
True