Coalgebras#
- class sage.categories.coalgebras.Coalgebras(base, name=None)#
Bases:
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]
- class Cocommutative(base_category)#
Bases:
CategoryWithAxiom_over_base_ring
Category of cocommutative coalgebras.
- class DualObjects(category, *args)#
Bases:
DualObjectsCategory
- extra_super_categories()#
Return 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]
Warning
This is only correct in certain cases (finite dimension, …). See github issue #15647.
- class ElementMethods#
Bases:
object
- coproduct()#
Return 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)])
- counit()#
Return the counit 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.counit() (B[(1,2,3)], 1) sage: b, b.counit() (B[(1,3)], 1)
- class Filtered(base_category)#
Bases:
FilteredModulesCategory
Category of filtered coalgebras.
- Graded#
alias of
GradedCoalgebras
- class ParentMethods#
Bases:
object
- coproduct(x)#
Return 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)])
- counit(x)#
Return the counit 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.counit(a) (B[(1,2,3)], 1) sage: b, A.counit(b) (B[(1,3)], 1)
TODO: implement some tests of the axioms of coalgebras, bialgebras and Hopf algebras using the counit.
- class Realizations(category, *args)#
Bases:
RealizationsCategory
- class ParentMethods#
Bases:
object
- coproduct_by_coercion(x)#
Return the coproduct by coercion if
coproduct_by_basis
is not implemented.EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: m = Sym.monomial() sage: f = m[2,1] sage: f.coproduct.__module__ 'sage.categories.coalgebras' sage: m.coproduct_on_basis NotImplemented sage: m.coproduct == m.coproduct_by_coercion True sage: f.coproduct() m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: R = N.ribbon() sage: R.coproduct_by_coercion.__module__ 'sage.categories.coalgebras' sage: R.coproduct_on_basis NotImplemented sage: R.coproduct == R.coproduct_by_coercion True sage: R[1].coproduct() R[] # R[1] + R[1] # R[]
- counit_by_coercion(x)#
Return the counit of
x
ifcounit_by_basis
is not implemented.EXAMPLES:
sage: sp = SymmetricFunctions(QQ).sp() sage: sp.an_element() 2*sp[] + 2*sp[1] + 3*sp[2] sage: sp.counit(sp.an_element()) 2 sage: o = SymmetricFunctions(QQ).o() sage: o.an_element() 2*o[] + 2*o[1] + 3*o[2] sage: o.counit(o.an_element()) -1
- class SubcategoryMethods#
Bases:
object
- Cocommutative()#
Return the full subcategory of the cocommutative objects of
self
.A coalgebra \(C\) is said to be cocommutative if
\[\Delta(c) = \sum_{(c)} c_{(1)} \otimes c_{(2)} = \sum_{(c)} c_{(2)} \otimes c_{(1)}\]in Sweedler’s notation for all \(c \in C\).
EXAMPLES:
sage: C1 = Coalgebras(ZZ).Cocommutative().WithBasis(); C1 Category of cocommutative coalgebras with basis over Integer Ring sage: C2 = Coalgebras(ZZ).WithBasis().Cocommutative() sage: C1 is C2 True sage: BialgebrasWithBasis(QQ).Cocommutative() Category of cocommutative bialgebras with basis over Rational Field
- class Super(base_category)#
Bases:
SuperModulesCategory
- class SubcategoryMethods#
Bases:
object
- Supercocommutative()#
Return the full subcategory of the supercocommutative objects of
self
.EXAMPLES:
sage: Coalgebras(ZZ).WithBasis().Super().Supercocommutative() Category of supercocommutative super coalgebras with basis over Integer Ring sage: BialgebrasWithBasis(QQ).Super().Supercocommutative() Join of Category of super algebras with basis over Rational Field and Category of super bialgebras over Rational Field and Category of super coalgebras with basis over Rational Field and Category of supercocommutative super coalgebras over Rational Field
- class Supercocommutative(base_category)#
Bases:
CategoryWithAxiom_over_base_ring
Category of supercocommutative coalgebras.
- extra_super_categories()#
EXAMPLES:
sage: Coalgebras(ZZ).Super().extra_super_categories() [Category of graded coalgebras over Integer Ring] sage: Coalgebras(ZZ).Super().super_categories() [Category of graded coalgebras over Integer Ring, Category of super modules over Integer Ring]
Compare this with the situation for bialgebras:
sage: Bialgebras(ZZ).Super().extra_super_categories() [] sage: Bialgebras(ZZ).Super().super_categories() [Category of super algebras over Integer Ring, Category of super coalgebras over Integer Ring]
The category of bialgebras does not occur in these results, since super bialgebras are not bialgebras.
- class TensorProducts(category, *args)#
Bases:
TensorProductsCategory
- class ElementMethods#
Bases:
object
- class ParentMethods#
Bases:
object
- extra_super_categories()#
EXAMPLES:
sage: Coalgebras(QQ).TensorProducts().extra_super_categories() [Category of coalgebras over Rational Field] sage: Coalgebras(QQ).TensorProducts().super_categories() [Category of tensor products of vector spaces over Rational Field, Category of coalgebras over Rational Field]
Meaning: a tensor product of coalgebras is a coalgebra
- WithBasis#
alias of
CoalgebrasWithBasis
- class WithRealizations(category, *args)#
Bases:
WithRealizationsCategory
- class ParentMethods#
Bases:
object
- coproduct(x)#
Return the coproduct of
x
.EXAMPLES:
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: S = N.complete() sage: N.coproduct.__module__ 'sage.categories.coalgebras' sage: N.coproduct(S[2]) S[] # S[2] + S[1] # S[1] + S[2] # S[]
- counit(x)#
Return the counit of
x
.EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.schur() sage: f = s[2,1] sage: f.counit.__module__ 'sage.categories.coalgebras' sage: f.counit() 0
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: N.counit.__module__ 'sage.categories.coalgebras' sage: N.counit(N.one()) 1 sage: x = N.an_element(); x 2*S[] + 2*S[1] + 3*S[1, 1] sage: N.counit(x) 2
- super_categories()#
EXAMPLES:
sage: Coalgebras(QQ).super_categories() [Category of vector spaces over Rational Field]