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


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
class ParentMethods

Return the associated graded algebra to self.

This is self, because self is already graded. See graded_algebra() for the general behavior of this method, and see AssociatedGradedAlgebra for the definition and properties of associated graded algebras.


sage: m = SymmetricFunctions(QQ).m()
sage: m.graded_algebra() is m
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

Implements operations on tensor products of super algebras with 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.


sage: A.<x,y> = ExteriorAlgebra(QQ)
sage: A.one_basis()
sage: B = tensor((A, A, A))
sage: B.one_basis()
((), (), ())
1 # 1 # 1
product_on_basis(t0, t1)

The product of the algebra on the basis, as per AlgebrasWithBasis.ParentMethods.product_on_basis.


Test the sign in the super tensor product:

sage: A = SteenrodAlgebra(3)
sage: x = A.Q(0)
sage: y = x.coproduct()
sage: y^2

TODO: optimize this implementation!



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]