HopfAlgebrasWithBasis

class sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of Hopf algebras with a distinguished basis

EXAMPLES:

sage: C = HopfAlgebrasWithBasis(QQ)
sage: C
Category of hopf algebras with basis over Rational Field
sage: C.super_categories()
[Category of bialgebras with basis over Rational Field, Category of hopf algebras over Rational Field]

We now show how to use a simple hopf algebra, namely the group algebra of the dihedral group (see also AlgebrasWithBasis):

sage: A = C.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.__custom_name = "A"
sage: A.category()
Category of hopf algebras with basis over Rational Field

sage: A.one_basis()
()
sage: A.one()
B[()]

sage: A.base_ring()
Rational Field
sage: A.basis().keys()
Dihedral group of order 6 as a permutation group

sage: [a,b] = A.algebra_generators()
sage: a, b
(B[(1,2,3)], B[(1,3)])
sage: a^3, b^2
(B[()], B[()])
sage: a*b
B[(1,2)]

sage: A.product           # todo: not quite ...
Generic endomorphism of A
sage: A.product(b,b)
B[()]

sage: A.zero().coproduct()
0
sage: A.zero().coproduct().parent()
A # A
sage: a.coproduct()
B[(1,2,3)] # B[(1,2,3)]

sage: TestSuite(A).run(verbose=True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_category() . . . pass
running ._test_distributivity() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq() . . . pass
running ._test_eq() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass
sage: A.__class__
<class 'sage.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category'>
sage: A.element_class
<class 'sage.combinat.free_module.MyGroupAlgebra_with_category.element_class'>

Let us look at the code for implementing A:

sage: A??                       # todo: not implemented

TESTS:

sage: TestSuite(A).run()
sage: TestSuite(C).run()

class ElementMethods

class HopfAlgebrasWithBasis.ParentMethods

antipode()

If antipode_basis() is available, construct the antipode morphism from self to self by extending it by linearity

EXAMPLES:

sage: A = HopfAlgebrasWithBasis(ZZ).example(); A
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Integer Ring
sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: [a,b] = A.algebra_generators()
sage: a, A.antipode(a)
(B[(1,2,3)], B[(1,3,2)])
sage: b, A.antipode(b)
(B[(1,3)], B[(1,3)])

TESTS:

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

class HopfAlgebrasWithBasis.TensorProducts(category, *args)

Bases: sage.categories.tensor.TensorProductsCategory

The category of hopf algebras with basis constructed by tensor product of hopf algebras with basis

ElementMethods

alias of TensorProducts.ElementMethods

ParentMethods

alias of TensorProducts.ParentMethods

extra_super_categories(*args, **kwds)

EXAMPLES:

sage: HopfAlgebrasWithBasis(QQ).TensorProducts().extra_super_categories()
[Category of hopf algebras with basis over Rational Field]
sage: HopfAlgebrasWithBasis(QQ).TensorProducts().super_categories()
[Category of hopf algebras with basis over Rational Field, Category of tensor products of algebras with basis over Rational Field, Category of tensor products of hopf algebras over Rational Field]

HopfAlgebrasWithBasis.example(G=None)

Returns an example of algebra with basis:

sage: HopfAlgebrasWithBasis(QQ[x]).example()
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Univariate Polynomial Ring in x over Rational Field

An other group can be specified as optional argument:

sage: HopfAlgebrasWithBasis(QQ).example(SymmetricGroup(4))
An example of Hopf algebra with basis: the group algebra of the SymmetricGroup(4) over Rational Field

HopfAlgebrasWithBasis.super_categories(*args, **kwds)

EXAMPLES:

sage: HopfAlgebrasWithBasis(QQ).super_categories()
[Category of bialgebras with basis over Rational Field, Category of hopf algebras over Rational Field]