Coalgebras with basis¶
- class sage.categories.coalgebras_with_basis.CoalgebrasWithBasis(base_category)¶
Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe 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
ncoproducts toself.Todo
Remove dependency on
modules_with_basismethods.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.FilteredModulesCategoryCategory of filtered coalgebras.
- class ParentMethods¶
Bases:
object- coproduct()¶
If
coproduct_on_basis()is available, construct the coproduct morphism fromselftoself\(\otimes\)selfby extending it by linearity. Otherwise, usecoproduct_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 ofself
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 fromselftoself\(\otimes\)selfby extending it by linearityEXAMPLES:
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 ofself
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]