Bases: sage.categories.category_types.Category_over_base_ring
The category of coalgebras with a distinguished basis
EXAMPLES:
sage: CoalgebrasWithBasis(ZZ)
Category of coalgebras with basis over Integer Ring
sage: CoalgebrasWithBasis(ZZ).super_categories()
[Category of modules with basis over Integer Ring, Category of coalgebras 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)
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)
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
EXAMPLES:
sage: CoalgebrasWithBasis(QQ).super_categories()
[Category of modules with basis over Rational Field, Category of coalgebras over Rational Field]