HopfAlgebras

class sage.categories.hopf_algebras.HopfAlgebras(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of Hopf algebras

EXAMPLES:

sage: HopfAlgebras(QQ)
Category of hopf algebras over Rational Field
sage: HopfAlgebras(QQ).super_categories()
[Category of bialgebras over Rational Field]

TESTS:

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

class DualCategory(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of Hopf algebras constructed as dual of a Hopf algebra

ParentMethods

alias of DualCategory.ParentMethods

class HopfAlgebras.ElementMethods

antipode()

Returns the antipode of self.

EXAMPLES:

sage: A = HopfAlgebrasWithBasis(QQ).example(); A
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
sage: [a,b] = A.algebra_generators()
sage: a, a.antipode()
(B[(1,2,3)], B[(1,3,2)])
sage: b, b.antipode()
(B[(1,3)], B[(1,3)])

TESTS:

sage: all(x.antipode() * x == A.one() for x in A.basis())
True

class HopfAlgebras.Morphism(s=None)

Bases: sage.categories.category.Category

The category of Hopf algebra morphisms

class HopfAlgebras.ParentMethods

class HopfAlgebras.TensorProducts(category, *args)

Bases: sage.categories.tensor.TensorProductsCategory

The category of Hopf algebras constructed by tensor product of Hopf algebras

ElementMethods

alias of TensorProducts.ElementMethods

ParentMethods

alias of TensorProducts.ParentMethods

extra_super_categories(*args, **kwds)

EXAMPLES:

sage: HopfAlgebras(QQ).TensorProducts().extra_super_categories()
[Category of hopf algebras over Rational Field]
sage: HopfAlgebras(QQ).TensorProducts().super_categories()
[Category of hopf algebras over Rational Field,
 Category of tensor products of algebras over Rational Field,
 Category of tensor products of coalgebras over Rational Field]

HopfAlgebras.dual()

Returns the dual category

EXAMPLES:

The category of Hopf algebras over any field is self dual:

sage: C = HopfAlgebras(QQ)
sage: C.dual()
Category of hopf algebras over Rational Field

HopfAlgebras.super_categories(*args, **kwds)

EXAMPLES:

sage: HopfAlgebras(QQ).super_categories()
[Category of bialgebras over Rational Field]