Coalgebras

class sage.categories.coalgebras.Coalgebras(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring, sage.categories.tensor.CategoryWithTensorProduct, sage.categories.dual.DualityCategory

The category of coalgebras

EXAMPLES:

sage: Coalgebras(QQ)
Category of coalgebras over Rational Field
sage: Coalgebras(QQ).super_categories()
[Category of modules over Rational Field]
sage: Coalgebras(QQ).all_super_categories() # todo: update once the hierarchy is more appropriate
[Category of coalgebras over Rational Field,
 Category of modules over Rational Field,
 Category of bimodules over Rational Field on the left and Rational Field on the right,
 Category of left modules over Rational Field,
 Category of right modules over Rational Field,
 Category of commutative additive groups,
 Category of commutative additive monoids,
 Category of commutative additive semigroups,
 Category of additive magmas,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

TESTS:

sage: TestSuite(Coalgebras(ZZ)).run()
class ElementMethods
coproduct()

Returns the coproduct 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.coproduct()
(B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
sage: b, b.coproduct()
(B[(1,3)], B[(1,3)] # B[(1,3)])
class Coalgebras.ParentMethods
coproduct(x)

Returns the coproduct of x.

Eventually, there will be a default implementation, delegating to the overloading mechanism and forcing the conversion back

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)])
tensor_square(*args, **kwds)

Returns the tensor square of self

EXAMPLES:

sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: A.tensor_square()
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field # An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
class Coalgebras.TensorCategory(category, name=None)

Bases: sage.categories.tensor.TensorCategory

ElementMethods
alias of TensorCategory.ElementMethods
ParentMethods
alias of TensorCategory.ParentMethods
super_categories(*args, **kwds)

EXAMPLES:

sage: Coalgebras(QQ).tensor_category().super_categories()
[Category of coalgebras over Rational Field]
Coalgebras.dual()

Returns the dual category

EXAMPLES:

The category of coalgebras over the Rational Field is dual to the category of algebras over the same field:

sage: C = Coalgebras(QQ)
sage: C.dual()
Category of algebras over Rational Field
Coalgebras.super_categories(*args, **kwds)

EXAMPLES:

sage: Coalgebras(QQ).super_categories()
[Category of modules over Rational Field]

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