Coalgebras with basis¶
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class
sage.categories.coalgebras_with_basis.
CoalgebrasWithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
The 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]
TESTS:
sage: TestSuite(CoalgebrasWithBasis(ZZ)).run()
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class
ElementMethods
¶ -
coproduct_iterated
(n=1)¶ Apply
n
coproducts toself
.Todo
Remove dependency on
modules_with_basis
methods.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[]
TESTS:
sage: p = SymmetricFunctions(QQ).p() sage: p[5,2,2].coproduct_iterated() p[] # p[5, 2, 2] + 2*p[2] # p[5, 2] + p[2, 2] # p[5] + p[5] # p[2, 2] + 2*p[5, 2] # p[2] + p[5, 2, 2] # p[] sage: p([]).coproduct_iterated(3) p[] # p[] # p[] # p[]
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() sage: Psi[2,2].coproduct_iterated(0) Psi[2, 2] sage: Psi[2,2].coproduct_iterated(3) Psi[] # Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[] # Psi[2] # Psi[2] + Psi[] # Psi[] # Psi[2, 2] # Psi[] + 2*Psi[] # Psi[2] # Psi[] # Psi[2] + 2*Psi[] # Psi[2] # Psi[2] # Psi[] + Psi[] # Psi[2, 2] # Psi[] # Psi[] + 2*Psi[2] # Psi[] # Psi[] # Psi[2] + 2*Psi[2] # Psi[] # Psi[2] # Psi[] + 2*Psi[2] # Psi[2] # Psi[] # Psi[] + Psi[2, 2] # Psi[] # Psi[] # Psi[]
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m() sage: m[[1,3],[2]].coproduct_iterated(2) m{} # m{} # m{{1, 3}, {2}} + m{} # m{{1}} # m{{1, 2}} + m{} # m{{1, 2}} # m{{1}} + m{} # m{{1, 3}, {2}} # m{} + m{{1}} # m{} # m{{1, 2}} + m{{1}} # m{{1, 2}} # m{} + m{{1, 2}} # m{} # m{{1}} + m{{1, 2}} # m{{1}} # m{} + m{{1, 3}, {2}} # m{} # m{} sage: m[[]].coproduct_iterated(3), m[[1,3],[2]].coproduct_iterated(0) (m{} # m{} # m{} # m{}, m{{1, 3}, {2}})
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class
CoalgebrasWithBasis.
ParentMethods
¶ -
coproduct
()¶ If
coproduct_on_basis()
is available, construct the coproduct morphism fromself
toself
\(\otimes\)self
by 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)])
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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)]
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counit
()¶ If
counit_on_basis()
is available, construct the counit morphism fromself
toself
\(\otimes\)self
by 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)
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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
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class
CoalgebrasWithBasis.
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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class