# Graded Hopf algebras with basis¶

class sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis(base_category)

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]

True
False

class Connected(base_category)
class ElementMethods

Bases: object

class ParentMethods

Bases: object

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 $$|x| = 0$$.

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

example()

Return an example of a graded connected Hopf algebra with a distinguished basis.

class ElementMethods

Bases: object

class ParentMethods

Bases: object

class WithRealizations(category, *args)
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
and Category of graded coalgebras over Rational Field]

example()

Return an example of a graded Hopf algebra with a distinguished basis.