Super Hopf algebras with basis

class sage.categories.super_hopf_algebras_with_basis.SuperHopfAlgebrasWithBasis(base_category)

Bases: SuperModulesCategory

The category of super Hopf algebras with a distinguished basis.

EXAMPLES:

Sage

sage: C = HopfAlgebras(ZZ).WithBasis().Super(); C
Category of super Hopf algebras with basis over Integer Ring
sage: sorted(C.super_categories(), key=str)
[Category of super Hopf algebras over Integer Ring,
 Category of super algebras with basis over Integer Ring,
 Category of super coalgebras with basis over Integer Ring]

Python

>>> from sage.all import *
>>> C = HopfAlgebras(ZZ).WithBasis().Super(); C
Category of super Hopf algebras with basis over Integer Ring
>>> sorted(C.super_categories(), key=str)
[Category of super Hopf algebras over Integer Ring,
 Category of super algebras with basis over Integer Ring,
 Category of super coalgebras with basis over Integer Ring]

class ParentMethods

Bases: object

antipode()

The antipode of this Hopf algebra.

If antipode_basis() is available, this constructs the antipode morphism from self to self by extending it by linearity. Otherwise, self.antipode_by_coercion() is used, if available.

EXAMPLES:

Sage

sage: A = SteenrodAlgebra(7)                                            # needs sage.combinat sage.modules
sage: a = A.an_element()                                                # needs sage.combinat sage.modules
sage: a, A.antipode(a)                                                  # needs sage.combinat sage.modules
(6 Q_1 Q_3 P(2,1), Q_1 Q_3 P(2,1))

Python

>>> from sage.all import *
>>> A = SteenrodAlgebra(Integer(7))                                            # needs sage.combinat sage.modules
>>> a = A.an_element()                                                # needs sage.combinat sage.modules
>>> a, A.antipode(a)                                                  # needs sage.combinat sage.modules
(6 Q_1 Q_3 P(2,1), Q_1 Q_3 P(2,1))