Hopf algebras with basis¶
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sage.categories.hopf_algebras_with_basis.
HopfAlgebrasWithBasis
¶ 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 hopf algebras over Rational Field, Category of bialgebras with basis 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 finite dimensional 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 ... <bound method MyGroupAlgebra_with_category._product_from_product_on_basis_multiply 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_antipode() . . . pass running ._test_associativity() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_characteristic() . . . 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_new() . . . pass running ._test_nonzero_equal() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_eq() . . . pass running ._test_new() . . . 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.categories.examples.hopf_algebras_with_basis.MyGroupAlgebra_with_category.element_class'>
Let us look at the code for implementing A:
sage: A?? # todo: not implemented