Graded Hopf algebras with basis¶
-
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
-
class
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
-
class
GradedHopfAlgebrasWithBasis.Connected.
ParentMethods
¶ -
antipode
(elem)¶
-
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 \(|x| = 0\).
-
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
-
-
GradedHopfAlgebrasWithBasis.Connected.
example
()¶
-
class
-
class
GradedHopfAlgebrasWithBasis.
ElementMethods
¶
-
class
GradedHopfAlgebrasWithBasis.
ParentMethods
¶
-
class
GradedHopfAlgebrasWithBasis.
WithRealizations
(category, *args)¶ Bases:
sage.categories.with_realizations.WithRealizationsCategory
-
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]
-
-
GradedHopfAlgebrasWithBasis.
example
()¶
-
class