Hopf algebras¶
- class sage.categories.hopf_algebras.HopfAlgebras(base, name=None)[source]¶
- Bases: - 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] - >>> 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 ElementMethods[source]¶
- Bases: - object- antipode()[source]¶
- Return the antipode of - self.- EXAMPLES: - 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)]) - >>> 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 Realizations(category, *args)[source]¶
- Bases: - RealizationsCategory- class ParentMethods[source]¶
- Bases: - object- antipode_by_coercion(x)[source]¶
- Return the image of - xby the antipode.- This default implementation coerces to the default realization, computes the antipode there, and coerces the result back. - EXAMPLES: - 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] - >>> 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: 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)) - >>> 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: C = HopfAlgebras(QQ).Super() sage: C.dual() Category of super Hopf algebras over Rational Field - >>> 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 - extra_super_categories()[source]¶
- 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] - >>> 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: C = HopfAlgebras(QQ) sage: C.dual() Category of Hopf algebras over Rational Field - >>> from sage.all import * >>> C = HopfAlgebras(QQ) >>> C.dual() Category of Hopf algebras over Rational Field