Graded algebras with basis¶

class
sage.categories.graded_algebras_with_basis.
GradedAlgebrasWithBasis
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
The category of graded algebras with a distinguished basis
EXAMPLES:
sage: C = GradedAlgebrasWithBasis(ZZ); C Category of graded algebras with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of filtered algebras with basis over Integer Ring, Category of graded algebras over Integer Ring, Category of graded modules with basis over Integer Ring]

class
ElementMethods
¶ Bases:
object

class
ParentMethods
¶ Bases:
object

graded_algebra
()¶ Return the associated graded algebra to
self
.This is
self
, becauseself
is already graded. Seegraded_algebra()
for the general behavior of this method, and seeAssociatedGradedAlgebra
for the definition and properties of associated graded algebras.EXAMPLES:
sage: m = SymmetricFunctions(QQ).m() sage: m.graded_algebra() is m True


class
SignedTensorProducts
(category, *args)¶ Bases:
sage.categories.signed_tensor.SignedTensorProductsCategory
The category of algebras with basis constructed by signed tensor product of algebras with basis.

class
ParentMethods
¶ Bases:
object
Implements operations on tensor products of super algebras with basis.

one_basis
()¶ Return the index of the one of this signed tensor product of algebras, as per
AlgebrasWithBasis.ParentMethods.one_basis
.It is the tuple whose operands are the indices of the ones of the operands, as returned by their
one_basis()
methods.EXAMPLES:
sage: A.<x,y> = ExteriorAlgebra(QQ) sage: A.one_basis() () sage: B = tensor((A, A, A)) sage: B.one_basis() ((), (), ()) sage: B.one() 1 # 1 # 1

product_on_basis
(t0, t1)¶ The product of the algebra on the basis, as per
AlgebrasWithBasis.ParentMethods.product_on_basis
.EXAMPLES:
Test the sign in the super tensor product:
sage: A = SteenrodAlgebra(3) sage: x = A.Q(0) sage: y = x.coproduct() sage: y^2 0
TODO: optimize this implementation!


extra_super_categories
()¶ EXAMPLES:
sage: Cat = AlgebrasWithBasis(QQ).Graded() sage: Cat.SignedTensorProducts().extra_super_categories() [Category of graded algebras with basis over Rational Field] sage: Cat.SignedTensorProducts().super_categories() [Category of graded algebras with basis over Rational Field, Category of signed tensor products of graded algebras over Rational Field]

class

class