Filtered algebra

Summary

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.

A filtered algebra over the field is an algebra over that has an increasing sequence of subspaces of such that

and that is compatible with the multiplication in the following sense:

Associated graded algebraEdit

In general there is the following construction that produces a graded algebra out of a filtered algebra.

If   is a filtered algebra then the associated graded algebra   is defined as follows:

  • As a vector space
     

    where,

      and
     
  • the multiplication is defined by
     

    for all   and  . (More precisely, the multiplication map   is combined from the maps

     
    for all   and  .)

The multiplication is well-defined and endows   with the structure of a graded algebra, with gradation   Furthermore if   is associative then so is  . Also if   is unital, such that the unit lies in  , then   will be unital as well.

As algebras   and   are distinct (with the exception of the trivial case that   is graded) but as vector spaces they are isomorphic. (One can prove by induction that   is isomorphic to   as vector spaces).

ExamplesEdit

Any graded algebra graded by  , for example  , has a filtration given by  .

An example of a filtered algebra is the Clifford algebra   of a vector space   endowed with a quadratic form   The associated graded algebra is  , the exterior algebra of  

The symmetric algebra on the dual of an affine space is a filtered algebra of polynomials; on a vector space, one instead obtains a graded algebra.

The universal enveloping algebra of a Lie algebra   is also naturally filtered. The PBW theorem states that the associated graded algebra is simply  .

Scalar differential operators on a manifold   form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded algebra is the commutative algebra of smooth functions on the cotangent bundle   which are polynomial along the fibers of the projection  .

The group algebra of a group with a length function is a filtered algebra.

See alsoEdit

ReferencesEdit

  • Abe, Eiichi (1980). Hopf Algebras. Cambridge: Cambridge University Press. ISBN 0-521-22240-0.

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.