Filtered Algebras

class sage.categories.filtered_algebras.FilteredAlgebras(base_category)

Bases: FilteredModulesCategory

The category of filtered algebras.

An algebra \(A\) over a commutative ring \(R\) is filtered if \(A\) is endowed with a structure of a filtered \(R\)-module (whose underlying \(R\)-module structure is identical with that of the \(R\)-algebra \(A\)) such that the indexing set \(I\) (typically \(I = \NN\)) is also an additive abelian monoid, the unity \(1\) of \(A\) belongs to \(F_0\), and we have \(F_i \cdot F_j \subseteq F_{i+j}\) for all \(i, j \in I\).

EXAMPLES:

Sage

sage: Algebras(ZZ).Filtered()
Category of filtered algebras over Integer Ring
sage: Algebras(ZZ).Filtered().super_categories()
[Category of algebras over Integer Ring,
 Category of filtered modules over Integer Ring]

Python

>>> from sage.all import *
>>> Algebras(ZZ).Filtered()
Category of filtered algebras over Integer Ring
>>> Algebras(ZZ).Filtered().super_categories()
[Category of algebras over Integer Ring,
 Category of filtered modules over Integer Ring]

REFERENCES:

• Wikipedia article Filtered_algebra

class ParentMethods

Bases: object

graded_algebra()

Return the associated graded algebra to self.

Todo

Implement a version of the associated graded algebra which does not require self to have a distinguished basis.

EXAMPLES:

Sage

sage: A = AlgebrasWithBasis(ZZ).Filtered().example()
sage: A.graded_algebra()
Graded Algebra of An example of a filtered algebra with basis:
 the universal enveloping algebra of
 Lie algebra of RR^3 with cross product over Integer Ring

Python

>>> from sage.all import *
>>> A = AlgebrasWithBasis(ZZ).Filtered().example()
>>> A.graded_algebra()
Graded Algebra of An example of a filtered algebra with basis:
 the universal enveloping algebra of
 Lie algebra of RR^3 with cross product over Integer Ring