Graded Hopf algebras with basis

class sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis(base_category)

Bases: sage.categories.graded_modules.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 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

TESTS:

sage: TestSuite(C).run()
class Connected(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

TESTS:

sage: C = Modules(ZZ).FiniteDimensional(); C
Category of finite dimensional modules over Integer Ring
sage: type(C)
<class 'sage.categories.modules.Modules.FiniteDimensional_with_category'>
sage: type(C).__base__.__base__
<class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'>

sage: TestSuite(C).run()
class ElementMethods
antipode()

TESTS:

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
sage: H.monomial(0).antipode()
P0
sage: H.monomial(2).antipode()
P2
sage: (2*H.monomial(1) + 3*H.monomial(4)).antipode()
-2*P1 + 3*P4
class GradedHopfAlgebrasWithBasis.Connected.ParentMethods
antipode(elem)

TESTS:

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
sage: H.antipode(H.monomial(140))
P140
antipode_on_basis(index)

The antipode on the basis element indexed by index.

INPUT:

  • index – an element of the index set
\[S(x) := -\sum_{x^L\neq x} S(x^L) \times x^R\]

in general or \(x\) if \(|x| = 0\).

TESTS:

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
sage: H.monomial(0).antipode() #indirect doctest
P0
sage: H.monomial(1).antipode() #indirect doctest
-P1
sage: H.monomial(2).antipode() #indirect doctest
P2
sage: H.monomial(3).antipode() #indirect doctest
-P3
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
\[\begin{split}c(i) := \begin{cases} 1 & \hbox{if $i$ is the unique element of degree $0$}\\ 0 & \hbox{otherwise}. \end{cases}\end{split}\]

EXAMPLES:

sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
sage: H.monomial(4).counit() # indirect doctest
0
sage: H.monomial(0).counit() # indirect doctest
1
GradedHopfAlgebrasWithBasis.Connected.example()

TESTS:

sage: GradedHopfAlgebrasWithBasis(QQ).Connected().example()
An example of a graded connected Hopf algebra with basis over Rational Field
class GradedHopfAlgebrasWithBasis.ElementMethods
class GradedHopfAlgebrasWithBasis.ParentMethods
class GradedHopfAlgebrasWithBasis.WithRealizations(category, *args)

Bases: sage.categories.with_realizations.WithRealizationsCategory

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
....:     _functor_category = "FooBars"
....:     _base_category_class = (Category,)
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
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]

TESTS:

sage: TestSuite(GradedHopfAlgebrasWithBasis(QQ).WithRealizations()).run()
GradedHopfAlgebrasWithBasis.example()

TESTS:

sage: GradedHopfAlgebrasWithBasis(QQ).example()
An example of a graded connected Hopf algebra with basis over Rational Field