# Hopf algebras¶

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

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)

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

class ParentMethods
class 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 Morphism(s=None)

The category of Hopf algebra morphisms.

class ParentMethods
class Realizations(category, *args)
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 Super(base_category)

The category of super Hopf algebras.

Note

A super Hopf algebra is not simply a Hopf algebra with a $$\ZZ/2\ZZ$$ grading due to the signed bialgebra compatibility conditions.

class ElementMethods
antipode()

Return the antipode of self.

EXAMPLES:

sage: A = SteenrodAlgebra(3)
sage: a = A.an_element()
sage: a, a.antipode()
(2 Q_1 Q_3 P(2,1), Q_1 Q_3 P(2,1))

dual()

Return the dual category.

EXAMPLES:

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

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

class TensorProducts(category, *args)

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

class ElementMethods
class ParentMethods
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]

WithBasis
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

super_categories()

EXAMPLES:

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