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]

TESTS:

sage: TestSuite(C).run()
class ElementMethods
class GradedAlgebrasWithBasis.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

TESTS:

Let us check that the three methods to_graded_conversion(), from_graded_conversion() and projection() (which form the interface of the associated graded algebra) work correctly here:

sage: to_gr = m.to_graded_conversion()
sage: from_gr = m.from_graded_conversion()
sage: m[2] == to_gr(m[2]) == from_gr(m[2])
True
sage: u = 3*m[1] - (1/2)*m[3]
sage: u == to_gr(u) == from_gr(u)
True
sage: m.zero() == to_gr(m.zero()) == from_gr(m.zero())
True
sage: p2 = m.projection(2)
sage: p2(m[2] - 4*m[1,1] + 3*m[1] - 2*m[[]])
-4*m[1, 1] + m[2]
sage: p2(4*m[1])
0
sage: p2(m.zero()) == m.zero()
True