Graded algebras with basis¶
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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()
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class
ElementMethods
¶
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class
GradedAlgebrasWithBasis.
ParentMethods
¶ -
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
TESTS:
Let us check that the three methods
to_graded_conversion()
,from_graded_conversion()
andprojection()
(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
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class