Super Lie Conformal Algebras

AUTHORS:

• Reimundo Heluani (2019-10-05): Initial implementation.

class sage.categories.super_lie_conformal_algebras.SuperLieConformalAlgebras(base_category)

Bases: SuperModulesCategory

The category of super Lie conformal algebras.

EXAMPLES:

Sage

sage: LieConformalAlgebras(AA).Super()                                          # needs sage.rings.number_field
Category of super Lie conformal algebras over Algebraic Real Field

Python

>>> from sage.all import *
>>> LieConformalAlgebras(AA).Super()                                          # needs sage.rings.number_field
Category of super Lie conformal algebras over Algebraic Real Field

Notice that we can force to have a purely even super Lie conformal algebra:

Sage

sage: bosondict = {('a','a'): {1:{('K',0):1}}}
sage: R = LieConformalAlgebra(QQ, bosondict, names=('a',),                      # needs sage.combinat sage.modules
....:                         central_elements=('K',), super=True)
sage: [g.is_even_odd() for g in R.gens()]                                       # needs sage.combinat sage.modules
[0, 0]

Python

>>> from sage.all import *
>>> bosondict = {('a','a'): {Integer(1):{('K',Integer(0)):Integer(1)}}}
>>> R = LieConformalAlgebra(QQ, bosondict, names=('a',),                      # needs sage.combinat sage.modules
...                         central_elements=('K',), super=True)
>>> [g.is_even_odd() for g in R.gens()]                                       # needs sage.combinat sage.modules
[0, 0]

class ElementMethods

Bases: object

is_even_odd()

Return 0 if this element is even and 1 if it is odd.

EXAMPLES:

Sage

sage: R = lie_conformal_algebras.NeveuSchwarz(QQ)                       # needs sage.combinat sage.modules
sage: R.inject_variables()                                              # needs sage.combinat sage.modules
Defining L, G, C
sage: G.is_even_odd()                                                   # needs sage.combinat sage.modules
1

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.NeveuSchwarz(QQ)                       # needs sage.combinat sage.modules
>>> R.inject_variables()                                              # needs sage.combinat sage.modules
Defining L, G, C
>>> G.is_even_odd()                                                   # needs sage.combinat sage.modules
1

class Graded(base_category)

Bases: GradedModulesCategory

The category of H-graded super Lie conformal algebras.

EXAMPLES:

Sage

sage: LieConformalAlgebras(AA).Super().Graded()                             # needs sage.rings.number_field
Category of H-graded super Lie conformal algebras over Algebraic Real Field

Python

>>> from sage.all import *
>>> LieConformalAlgebras(AA).Super().Graded()                             # needs sage.rings.number_field
Category of H-graded super Lie conformal algebras over Algebraic Real Field

class ParentMethods

Bases: object

example()

An example parent in this category.

EXAMPLES:

Sage

sage: LieConformalAlgebras(QQ).Super().example()                            # needs sage.combinat sage.modules
The Neveu-Schwarz super Lie conformal algebra over Rational Field

Python

>>> from sage.all import *
>>> LieConformalAlgebras(QQ).Super().example()                            # needs sage.combinat sage.modules
The Neveu-Schwarz super Lie conformal algebra over Rational Field

extra_super_categories()

The extra super categories of self.

EXAMPLES:

Sage

sage: LieConformalAlgebras(QQ).Super().super_categories()
[Category of super modules over Rational Field,
 Category of Lambda bracket algebras over Rational Field]

Python

>>> from sage.all import *
>>> LieConformalAlgebras(QQ).Super().super_categories()
[Category of super modules over Rational Field,
 Category of Lambda bracket algebras over Rational Field]