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# 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]

  TESTS:

      sage: TestSuite(HopfAlgebras(ZZ)).run()

  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`()

      Returns 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)])

      TESTS:

          sage: all(x.antipode() * x == A.one() for x in A.basis())
          True

  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`

    TESTS:

        sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
        sage: class FooBars(CovariantConstructionCategory):
        ...       _functor_category = "FooBars"
        sage: Category.FooBars = lambda self: FooBars.category_of(self)
        sage: C = FooBars(ModulesWithBasis(ZZ))
        sage: C
        Category of foo bars of modules with basis over Integer Ring
        sage: C.base_category()
        Category of modules with basis over Integer Ring
        sage: latex(C)
        \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
        sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
        sage: TestSuite(C).run()

    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.``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`()

    Returns 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]

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