Filtered Algebras - Category Framework

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# 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