Graded Hopf algebras with basis¶
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class
sage.categories.graded_hopf_algebras_with_basis.
GradedHopfAlgebrasWithBasis
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
The category of graded Hopf algebras with a distinguished basis.
EXAMPLES:
sage: C = GradedHopfAlgebrasWithBasis(ZZ); C Category of graded hopf algebras with basis over Integer Ring sage: C.super_categories() [Category of hopf algebras with basis over Integer Ring, Category of graded algebras with basis over Integer Ring] sage: C is HopfAlgebras(ZZ).WithBasis().Graded() True sage: C is HopfAlgebras(ZZ).Graded().WithBasis() False
TESTS:
sage: TestSuite(C).run()
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class
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
TESTS:
sage: C = Modules(ZZ).FiniteDimensional(); C Category of finite dimensional modules over Integer Ring sage: type(C) <class 'sage.categories.modules.Modules.FiniteDimensional_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'> sage: TestSuite(C).run()
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class
ElementMethods
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antipode
()¶ TESTS:
sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() sage: H.monomial(0).antipode() P0 sage: H.monomial(2).antipode() P2 sage: (2*H.monomial(1) + 3*H.monomial(4)).antipode() -2*P1 + 3*P4
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class
GradedHopfAlgebrasWithBasis.Connected.
ParentMethods
¶ -
antipode
(elem)¶ TESTS:
sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() sage: H.antipode(H.monomial(140)) P140
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antipode_on_basis
(index)¶ The antipode on the basis element indexed by
index
.INPUT:
index
– an element of the index set
\[S(x) := -\sum_{x^L\neq x} S(x^L) \times x^R\]in general or \(x\) if \(|x| = 0\).
TESTS:
sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() sage: H.monomial(0).antipode() #indirect doctest P0 sage: H.monomial(1).antipode() #indirect doctest -P1 sage: H.monomial(2).antipode() #indirect doctest P2 sage: H.monomial(3).antipode() #indirect doctest -P3
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counit_on_basis
(i)¶ The default counit of a graded connected Hopf algebra.
INPUT:
i
– an element of the index set
OUTPUT:
- an element of the base ring
\[\begin{split}c(i) := \begin{cases} 1 & \hbox{if $i$ is the unique element of degree $0$}\\ 0 & \hbox{otherwise}. \end{cases}\end{split}\]EXAMPLES:
sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example() sage: H.monomial(4).counit() # indirect doctest 0 sage: H.monomial(0).counit() # indirect doctest 1
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GradedHopfAlgebrasWithBasis.Connected.
example
()¶ TESTS:
sage: GradedHopfAlgebrasWithBasis(QQ).Connected().example() An example of a graded connected Hopf algebra with basis over Rational Field
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class
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class
GradedHopfAlgebrasWithBasis.
ElementMethods
¶
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class
GradedHopfAlgebrasWithBasis.
ParentMethods
¶
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class
GradedHopfAlgebrasWithBasis.
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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super_categories
()¶ EXAMPLES:
sage: GradedHopfAlgebrasWithBasis(QQ).WithRealizations().super_categories() [Join of Category of hopf algebras over Rational Field and Category of graded algebras over Rational Field]
TESTS:
sage: TestSuite(GradedHopfAlgebrasWithBasis(QQ).WithRealizations()).run()
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GradedHopfAlgebrasWithBasis.
example
()¶ TESTS:
sage: GradedHopfAlgebrasWithBasis(QQ).example() An example of a graded connected Hopf algebra with basis over Rational Field
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class