Graded Hopf algebras with basis¶
- class sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis(base_category)[source]¶
- Bases: - GradedModulesCategory- The category of graded Hopf algebras with a distinguished basis. - EXAMPLES: - sage: C = GradedHopfAlgebrasWithBasis(ZZ); C Category of graded Hopf algebras with basis over Integer Ring sage: C.super_categories() [Category of filtered Hopf algebras with basis over Integer Ring, Category of graded algebras with basis over Integer Ring, Category of graded coalgebras with basis over Integer Ring] sage: C is HopfAlgebras(ZZ).WithBasis().Graded() True sage: C is HopfAlgebras(ZZ).Graded().WithBasis() False - >>> from sage.all import * >>> C = GradedHopfAlgebrasWithBasis(ZZ); C Category of graded Hopf algebras with basis over Integer Ring >>> C.super_categories() [Category of filtered Hopf algebras with basis over Integer Ring, Category of graded algebras with basis over Integer Ring, Category of graded coalgebras with basis over Integer Ring] >>> C is HopfAlgebras(ZZ).WithBasis().Graded() True >>> C is HopfAlgebras(ZZ).Graded().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 graded connected Hopf algebra, we can define an antipode recursively by \[S(x) := -\sum_{x^L \neq x} S(x^L) \times x^R\]- when \(|x| > 0\), and by \(S(x) = x\) when \(|x| = 0\). 
 - counit_on_basis(i)[source]¶
- The default counit of a graded connected Hopf algebra. - INPUT: - i– an element of the index set
 - OUTPUT: an element of the base ring \[\begin{split}c(i) := \begin{cases} 1 & \hbox{if $i$ indexes the $1$ of the algebra}\\ 0 & \hbox{otherwise}. \end{cases}\end{split}\]- EXAMPLES: - sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() # needs sage.modules sage: H.monomial(4).counit() # indirect doctest # needs sage.modules 0 sage: H.monomial(0).counit() # indirect doctest # needs sage.modules 1 - >>> from sage.all import * >>> H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() # needs sage.modules >>> H.monomial(Integer(4)).counit() # indirect doctest # needs sage.modules 0 >>> H.monomial(Integer(0)).counit() # indirect doctest # needs sage.modules 1 
 
 
 - class WithRealizations(category, *args)[source]¶
- Bases: - WithRealizationsCategory- super_categories()[source]¶
- EXAMPLES: - sage: GradedHopfAlgebrasWithBasis(QQ).WithRealizations().super_categories() [Join of Category of Hopf algebras over Rational Field and Category of graded algebras over Rational Field and Category of graded coalgebras over Rational Field] - >>> from sage.all import * >>> GradedHopfAlgebrasWithBasis(QQ).WithRealizations().super_categories() [Join of Category of Hopf algebras over Rational Field and Category of graded algebras over Rational Field and Category of graded coalgebras over Rational Field]