Hopf algebras

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

class 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.Realizations(category, *args)

Bases: sage.categories.realizations.RealizationsCategory

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()
class ParentMethods
antipode_by_coercion(x)

Returns the image of x by the antipode

This default implementation coerces to the default realization, computes the antipode there, and coerces the result back.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: R.antipode_by_coercion.__module__
'sage.categories.hopf_algebras'
sage: R.antipode_by_coercion(R[1,3,1])
-R[2, 1, 2]
class HopfAlgebras.TensorProducts(category, *args)

Bases: sage.categories.tensor.TensorProductsCategory

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

class ElementMethods
class HopfAlgebras.TensorProducts.ParentMethods
HopfAlgebras.TensorProducts.extra_super_categories()

EXAMPLES:

sage: C = HopfAlgebras(QQ).TensorProducts()
sage: C.extra_super_categories()
[Category of hopf algebras over Rational Field]
sage: sorted(C.super_categories(), key=str)
[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.WithBasis

alias of HopfAlgebrasWithBasis

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()

EXAMPLES:

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

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