Hopf algebras¶
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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]
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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
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class
ParentMethods
¶ Bases:
object
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class
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class
ElementMethods
¶ Bases:
object
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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)])
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class
Morphism
(s=None)¶ Bases:
sage.categories.category.Category
The category of Hopf algebra morphisms.
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class
ParentMethods
¶ Bases:
object
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class
Realizations
(category, *args)¶ Bases:
sage.categories.realizations.RealizationsCategory
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class
ParentMethods
¶ Bases:
object
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antipode_by_coercion
(x)¶ Returns the image of
x
by the antipodeThis 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]
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class
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class
Super
(base_category)¶ Bases:
sage.categories.super_modules.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.
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class
ElementMethods
¶ Bases:
object
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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))
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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
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class
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class
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
The category of Hopf algebras constructed by tensor product of Hopf algebras
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class
ElementMethods
¶ Bases:
object
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class
ParentMethods
¶ Bases:
object
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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]
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class
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WithBasis
¶ alias of
sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis
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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
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super_categories
()¶ EXAMPLES:
sage: HopfAlgebras(QQ).super_categories() [Category of bialgebras over Rational Field]
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class