Super Hopf algebras with basis¶
- class sage.categories.super_hopf_algebras_with_basis.SuperHopfAlgebrasWithBasis(base_category)[source]¶
- Bases: - SuperModulesCategory- The category of super Hopf algebras with a distinguished basis. - EXAMPLES: - 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] - >>> 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[source]¶
- Bases: - object- antipode()[source]¶
- The antipode of this Hopf algebra. - If - antipode_basis()is available, this constructs the antipode morphism from- selfto- selfby extending it by linearity. Otherwise,- self.antipode_by_coercion()is used, if available.- EXAMPLES: - 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)) - >>> 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))