# Graded Hopf algebras with basis¶

The category of graded Hopf algebras with a distinguished basis

EXAMPLES:

sage: GradedHopfAlgebrasWithBasis(ZZ)
Category of graded hopf algebras with basis over Integer Ring
[Category of graded bialgebras with basis over Integer Ring, Category of graded hopf algebras over Integer Ring, Category of hopf algebras with basis over Integer Ring]


TESTS:

sage: TestSuite(GradedHopfAlgebrasWithBasis(ZZ)).run()

class ElementMethods

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
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()
[Category of graded hopf algebras over Rational Field]


TESTS:

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


EXAMPLES:

sage: GradedHopfAlgebrasWithBasis(QQ).super_categories()
[Category of graded bialgebras with basis over Rational Field, Category of graded hopf algebras over Rational Field, Category of hopf algebras with basis over Rational Field]