Coalgebras

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

Bases: sage.categories.category_types.Category_over_base_ring

The category of coalgebras

EXAMPLES:

sage: Coalgebras(QQ)
Category of coalgebras over Rational Field
sage: Coalgebras(QQ).super_categories()
[Category of vector spaces over Rational Field]
sage: Coalgebras(QQ).all_super_categories()
[Category of coalgebras over Rational Field,
 Category of vector spaces 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 DualObjects(category, *args)

Bases: sage.categories.dual.DualObjectsCategory

extra_super_categories()

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 duals of coalgebras over Rational Field
sage: C.dual().super_categories() # indirect doctest
[Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]

class Coalgebras.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.TensorProducts(category, *args)

Bases: sage.categories.tensor.TensorProductsCategory

ElementMethods

alias of TensorProducts.ElementMethods

ParentMethods

alias of TensorProducts.ParentMethods

extra_super_categories(*args, **kwds)

EXAMPLES:

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

Meaning: a tensor product of coalgebras is a coalgebra

Coalgebras.super_categories(*args, **kwds)

EXAMPLES:

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