Hopf algebras#

class sage.categories.hopf_algebras.HopfAlgebras(base, name=None)[source]#

Bases: Category_over_base_ring

The category of Hopf algebras.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> HopfAlgebras(QQ)
Category of Hopf algebras over Rational Field
>>> HopfAlgebras(QQ).super_categories()
[Category of bialgebras over Rational Field]

class DualCategory(base, name=None)[source]#

Bases: Category_over_base_ring

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

class ParentMethods[source]#: Bases: object

class ElementMethods[source]#

Bases: object

antipode()[source]#

Return the antipode of self

EXAMPLES:

Sage

sage: # needs sage.groups
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)])

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> 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
>>> [a,b] = A.algebra_generators()
>>> a, a.antipode()
(B[(1,2,3)], B[(1,3,2)])
>>> b, b.antipode()
(B[(1,3)], B[(1,3)])

class Morphism[source]#

Bases: Category

The category of Hopf algebra morphisms.

class ParentMethods[source]#: Bases: object

class Realizations(category, *args)[source]#

Bases: RealizationsCategory

class ParentMethods[source]#

Bases: object

antipode_by_coercion(x)[source]#

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

sage: # needs sage.combinat sage.modules
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]

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> N = NonCommutativeSymmetricFunctions(QQ)
>>> R = N.ribbon()
>>> R.antipode_by_coercion.__module__
'sage.categories.hopf_algebras'
>>> R.antipode_by_coercion(R[Integer(1),Integer(3),Integer(1)])
-R[2, 1, 2]

class Super(base_category)[source]#

Bases: SuperModulesCategory

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[source]#

Bases: object

antipode()[source]#

Return the antipode of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> A = SteenrodAlgebra(Integer(3))                                        # needs sage.combinat sage.modules
>>> a = A.an_element()                                            # needs sage.combinat sage.modules
>>> a, a.antipode()                                               # needs sage.combinat sage.modules
(2 Q_1 Q_3 P(2,1), Q_1 Q_3 P(2,1))

dual()[source]#

Return the dual category.

EXAMPLES:

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

Sage

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

Python

>>> from sage.all import *
>>> C = HopfAlgebras(QQ).Super()
>>> C.dual()
Category of super Hopf algebras over Rational Field

class TensorProducts(category, *args)[source]#

Bases: TensorProductsCategory

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

class ElementMethods[source]#: Bases: object

class ParentMethods[source]#: Bases: object

extra_super_categories()[source]#

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> C = HopfAlgebras(QQ).TensorProducts()
>>> C.extra_super_categories()
[Category of Hopf algebras over Rational Field]
>>> 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[source]#: alias of HopfAlgebrasWithBasis

dual()[source]#

Return the dual category

EXAMPLES:

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

Sage

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

Python

>>> from sage.all import *
>>> C = HopfAlgebras(QQ)
>>> C.dual()
Category of Hopf algebras over Rational Field

super_categories()[source]#

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> HopfAlgebras(QQ).super_categories()
[Category of bialgebras over Rational Field]