Filtered Hopf algebras with basis¶
- class sage.categories.filtered_hopf_algebras_with_basis.FilteredHopfAlgebrasWithBasis(base_category)[source]¶
Bases:
FilteredModulesCategoryThe 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
FilteredModulesWithBasisfor 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 ParentMethods[source]¶
Bases:
object- 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]