Coalgebras with basis#

class sage.categories.coalgebras_with_basis.CoalgebrasWithBasis(base_category)#

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]

class ElementMethods#

Bases: object

coproduct_iterated(n=1)#

Apply n coproducts to self.

Todo

Remove dependency on modules_with_basis methods.

EXAMPLES:

sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,2].coproduct_iterated(0)
Psi[2, 2]
sage: Psi[2,2].coproduct_iterated(2)
Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[2] # Psi[2]
 + Psi[] # Psi[2, 2] # Psi[] + 2*Psi[2] # Psi[] # Psi[2]
 + 2*Psi[2] # Psi[2] # Psi[] + Psi[2, 2] # Psi[] # Psi[]

class Filtered(base_category)#

Bases: sage.categories.filtered_modules.FilteredModulesCategory

Category of filtered coalgebras.

Graded#: alias of sage.categories.graded_coalgebras_with_basis.GradedCoalgebrasWithBasis

class ParentMethods#

Bases: object

coproduct()#

If coproduct_on_basis() is available, construct the coproduct morphism from self to self \(\otimes\) 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)])

coproduct_on_basis(i)#

The coproduct of the algebra on the basis (optional).

INPUT:

i – the indices of an element of the basis of self

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)]

counit()#

If counit_on_basis() is available, construct the counit morphism from self to self \(\otimes\) 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)

counit_on_basis(i)#

The counit of the algebra on the basis (optional).

INPUT:

i – the indices of an element of the basis of self

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

class Super(base_category)#

Bases: sage.categories.super_modules.SuperModulesCategory

extra_super_categories()#

EXAMPLES:

sage: C = Coalgebras(ZZ).WithBasis().Super()
sage: sorted(C.super_categories(), key=str)  # indirect doctest
[Category of graded coalgebras with basis over Integer Ring,
 Category of super coalgebras over Integer Ring,
 Category of super modules with basis over Integer Ring]