filtered algebra

Definition 1.

A filtered algebra over the field k is an algebra (A,) over k which is endowed with a filtrationMathworldPlanetmathPlanetmath ={Fi}i by subspacesPlanetmathPlanetmath, compatibleMathworldPlanetmath with the multiplicationPlanetmathPlanetmath in the following sense


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

Definition 2.

Let (A,,) be a filtered algebra then the associated algebra 𝒢(A) is defined as follows:

  • As a vector spaceMathworldPlanetmath



    G0=F0,and n>0,Gn=Fn/Fn-1,
  • the multiplication is defined by

Theorem 3.

The multiplication is well defined and endows G(A) with the of a graded algebra, with gradation {Gn}nN. Furthermore if A is associative then so is G(A).

An example of a filtered algebra is the Clifford algebraMathworldPlanetmathPlanetmath Cliff(V,q) of a vector space V endowed with a quadratic formMathworldPlanetmath q. The associated graded algebra is V, the exterior algebraMathworldPlanetmath of V.

As algebras A and 𝒢(A) are distinct (with the exception of the trivial case that A is graded) but as vector spaces they are isomorphic.

Theorem 4.

The underlying vector spaces of A and G(A) are isomorphic.

Title filtered algebra
Canonical name FilteredAlgebra
Date of creation 2013-03-22 13:23:55
Last modified on 2013-03-22 13:23:55
Owner Dr_Absentius (537)
Last modified by Dr_Absentius (537)
Numerical id 11
Author Dr_Absentius (537)
Entry type Definition
Classification msc 08A99