Coalgebras¶
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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]
TESTS:
sage: TestSuite(Coalgebras(ZZ)).run()
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class
DualObjects
(category, *args)¶ Bases:
sage.categories.dual.DualObjectsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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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 trac ticket #15647.
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class
Coalgebras.
ElementMethods
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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)])
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counit
()¶ Returns 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)
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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)])
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counit
(x)¶ Returns 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.
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class
Coalgebras.
Realizations
(category, *args)¶ Bases:
sage.categories.realizations.RealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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class
ParentMethods
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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[]
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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
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class
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class
Coalgebras.
Super
(base_category)¶ Bases:
sage.categories.super_modules.SuperModulesCategory
EXAMPLES:
sage: C = Algebras(QQ).Super() sage: C Category of super algebras over Rational Field sage: C.base_category() Category of algebras over Rational Field sage: sorted(C.super_categories(), key=str) [Category of graded algebras over Rational Field, Category of super modules over Rational Field] sage: AlgebrasWithBasis(QQ).Super().base_ring() Rational Field sage: HopfAlgebrasWithBasis(QQ).Super().base_ring() Rational Field
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extra_super_categories
()¶ EXAMPLES:
sage: Coalgebras(ZZ).Super().extra_super_categories() [Join of Category of graded modules over Integer Ring and Category of coalgebras over Integer Ring] sage: Coalgebras(ZZ).Super().super_categories() [Category of super modules over Integer Ring, Category of coalgebras 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.
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class
Coalgebras.
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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class
ElementMethods
¶
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class
Coalgebras.TensorProducts.
ParentMethods
¶
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Coalgebras.TensorProducts.
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
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class
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Coalgebras.
WithBasis
¶ alias of
CoalgebrasWithBasis
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class
Coalgebras.
WithRealizations
(category, *args)¶ Bases:
sage.categories.with_realizations.WithRealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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class
ParentMethods
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coproduct
(x)¶ Returns 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[]
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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
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class
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Coalgebras.
super_categories
()¶ EXAMPLES:
sage: Coalgebras(QQ).super_categories() [Category of vector spaces over Rational Field]
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class