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
class ParentMethods
graded_algebra()

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.

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

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]