Graded algebras with basis#
- class sage.categories.graded_algebras_with_basis.GradedAlgebrasWithBasis(base_category)#
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 completion of all formal linear combinations of
selfwith finite linear combinations in each homogeneous degree (computed lazily).
sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: S = NCSF.Complete() sage: L = S.formal_series_ring() sage: L Lazy completion of Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
- free_graded_module(generator_degrees, names=None)#
Create a finitely generated free graded module over
generator_degrees– tuple of integers defining the number of generators of the module and their degrees
names– (optional) the names of the generators. If
namesis a comma-separated string like
'a, b, c', then those will be the names. Otherwise, for example if
abc, then the names will be
By default, if all generators are in distinct degrees, then the
namesof the generators will have the form
dis the degree of the generator. If the degrees are not distinct, then the generators will be called
dis the degree and
iis its index in the list of generators in that degree.
sage.modules.fp_graded.free_modulefor more examples and details.
sage: Q = QuadraticForm(QQ, 3, [1,2,3,4,5,6]) sage: Cl = CliffordAlgebra(Q) sage: M = Cl.free_graded_module((0, 2, 3)) sage: M.gens() (g, g, g) sage: N.<xy, z> = Cl.free_graded_module((1, 2)) sage: N.generators() (xy, z)
Return the associated graded algebra to
selfis already graded. See
graded_algebra()for the general behavior of this method, and see
AssociatedGradedAlgebrafor the definition and properties of associated graded algebras.
sage: m = SymmetricFunctions(QQ).m() sage: m.graded_algebra() is m True
- class SignedTensorProducts(category, *args)#
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
It is the tuple whose operands are the indices of the ones of the operands, as returned by their
sage: A.<x,y> = ExteriorAlgebra(QQ) sage: A.one_basis() 0 sage: B = tensor((A, A, A)) sage: B.one_basis() (0, 0, 0) sage: B.one() 1 # 1 # 1
- product_on_basis(t0, t1)#
The product of the algebra on the basis, as per
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!
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]