Graded Hopf algebras with basis

class sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis(base_category)

Bases: GradedModulesCategory

The category of graded Hopf algebras with a distinguished basis.

EXAMPLES:

Sage

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

Python

>>> 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)

Bases: CategoryWithAxiom_over_base_ring

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

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

Python

>>> 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

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)

Bases: WithRealizationsCategory

super_categories()

EXAMPLES:

Sage

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]

Python

>>> 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]

example()

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