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]

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()

Return 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)])

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

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.Super(base_category)

Bases: sage.categories.super_modules.SuperModulesCategory

EXAMPLES:

sage: C = Algebras(QQ).Super()
sage: C
Category of super algebras over Rational Field
sage: C.base_category()
Category of algebras over Rational Field
sage: sorted(C.super_categories(), key=str)
[Category of graded algebras over Rational Field,
 Category of super modules over Rational Field]

sage: AlgebrasWithBasis(QQ).Super().base_ring()
Rational Field
sage: HopfAlgebrasWithBasis(QQ).Super().base_ring()
Rational Field

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()

Return 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]