Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
The category of coalgebras with a distinguished basis.
EXAMPLES:
sage: CoalgebrasWithBasis(ZZ)
Category of coalgebras with basis over Integer Ring
sage: sorted(CoalgebrasWithBasis(ZZ).super_categories(), key=str)
[Category of coalgebras over Integer Ring,
Category of modules with basis over Integer Ring]
TESTS:
sage: TestSuite(CoalgebrasWithBasis(ZZ)).run()
If coproduct_on_basis() is available, construct the
coproduct morphism from self to self
self by extending it by linearity. Otherwise, use
coproduct_by_coercion(),
if available.
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.coproduct(a)
(B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
sage: b, A.coproduct(b)
(B[(1,3)], B[(1,3)] # B[(1,3)])
The coproduct of the algebra on the basis (optional).
INPUT:
Returns the coproduct of the corresponding basis elements If implemented, the coproduct of the algebra is defined from it by linearity.
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._group.gens()
sage: A.coproduct_on_basis(a)
B[(1,2,3)] # B[(1,2,3)]
If counit_on_basis() is available, construct the
counit morphism from self to self
self by extending it by linearity
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.counit(a)
(B[(1,2,3)], 1)
sage: b, A.counit(b)
(B[(1,3)], 1)
The counit of the algebra on the basis (optional).
INPUT:
Returns the counit of the corresponding basis elements If implemented, the counit of the algebra is defined from it by linearity.
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._group.gens()
sage: A.counit_on_basis(a)
1