Filtered Hopf algebras with basis

class sage.categories.filtered_hopf_algebras_with_basis.FilteredHopfAlgebrasWithBasis(base_category)[source]

Bases: FilteredModulesCategory

The category of filtered Hopf algebras with a distinguished basis.

A filtered Hopf algebra with basis over a commutative ring \(R\) is a filtered Hopf algebra over \(R\) (that is, a Hopf algebra equipped with a module filtration such that all structure maps of the Hopf algebra respect the filtration) equipped with an \(R\)-module basis that makes it a filtered \(R\)-module with basis (see FilteredModulesWithBasis for the notion of a filtered module with basis).

EXAMPLES:

sage: C = HopfAlgebrasWithBasis(ZZ).Filtered(); C
Category of filtered Hopf algebras with basis over Integer Ring
sage: C.super_categories()
[Category of Hopf algebras with basis over Integer Ring,
 Category of filtered algebras with basis over Integer Ring,
 Category of filtered coalgebras with basis over Integer Ring]

sage: C is HopfAlgebras(ZZ).WithBasis().Filtered()
True
sage: C is HopfAlgebras(ZZ).Filtered().WithBasis()
False
>>> from sage.all import *
>>> C = HopfAlgebrasWithBasis(ZZ).Filtered(); C
Category of filtered Hopf algebras with basis over Integer Ring
>>> C.super_categories()
[Category of Hopf algebras with basis over Integer Ring,
 Category of filtered algebras with basis over Integer Ring,
 Category of filtered coalgebras with basis over Integer Ring]

>>> C is HopfAlgebras(ZZ).WithBasis().Filtered()
True
>>> C is HopfAlgebras(ZZ).Filtered().WithBasis()
False
class Connected(base_category)[source]

Bases: CategoryWithAxiom_over_base_ring

class ElementMethods[source]

Bases: object

class ParentMethods[source]

Bases: object

antipode(elem)[source]

Return the antipode of self applied to elem.

antipode_on_basis(index)[source]

The antipode on the basis element indexed by index.

INPUT:

  • index – an element of the index set

For a filtered connected Hopf algebra, we can define an antipode recursively by

\[S(x) := -\sum_{x^L \neq x} S(x^L) \times x^R + \epsilon(x)\]

for all \(x\), using the Sweedler notation. Also, \(S(x) = x\) for all \(x\) with \(|x| = 0\).

class WithRealizations(category, *args)[source]

Bases: WithRealizationsCategory

super_categories()[source]

EXAMPLES:

sage: HopfAlgebrasWithBasis(QQ).Filtered().WithRealizations().super_categories()
[Join of Category of Hopf algebras over Rational Field
     and Category of filtered algebras over Rational Field
     and Category of filtered coalgebras over Rational Field]
>>> from sage.all import *
>>> HopfAlgebrasWithBasis(QQ).Filtered().WithRealizations().super_categories()
[Join of Category of Hopf algebras over Rational Field
     and Category of filtered algebras over Rational Field
     and Category of filtered coalgebras over Rational Field]