Graded Hopf algebras with basis¶
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class
sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategoryThe 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] sage: C is HopfAlgebras(ZZ).WithBasis().Graded() True sage: C is HopfAlgebras(ZZ).Graded().WithBasis() False
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class
Connected(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring-
class
ElementMethods¶
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ParentMethods¶ -
antipode_on_basis(index)¶ 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 \(|x| = 0\).
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counit_on_basis(i)¶ 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() sage: H.monomial(4).counit() # indirect doctest 0 sage: H.monomial(0).counit() # indirect doctest 1
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example()¶ Return an example of a graded connected Hopf algebra with a distinguished basis.
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ElementMethods¶
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ParentMethods¶
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class
WithRealizations(category, *args)¶ Bases:
sage.categories.with_realizations.WithRealizationsCategory-
super_categories()¶ EXAMPLES:
sage: GradedHopfAlgebrasWithBasis(QQ).WithRealizations().super_categories() [Join of Category of hopf algebras over Rational Field and Category of graded algebras over Rational Field]
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example()¶ Return an example of a graded Hopf algebra with a distinguished basis.
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