Filtered algebra
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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 k is an algebra 
over k which is endowed with a filtration
compatible with the multiplication in the following sense
A special case of filtered algebra is a graded algebra.
[edit] Associated graded
In general there is the following construction that produces a graded algebra out of a filtered algebra.
If 
as a filtered algebra then the associated graded algebra 
is defined as follows:
- As a vector space
- G0 = F0, and
- the multiplication is defined by
The multiplication is well defined and endows with the structure of a graded algebra, with gradation 
Furthermore if A is associative then so is 
As algebras A and are distinct (with the exception of the trivial case that A is graded) but as vector spaces they are isomorphic.
[edit] Examples
An example of a filtered algebra is the Clifford algebra Cliff(V,q) of a vector space V endowed with a quadratic form q. The associated graded algebra is 
, the exterior algebra of V.
The universal enveloping algebra of a Lie algebra 
is also naturally filtered. The PBW theorem states that the associated graded algebra is simply 
.
This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the GFDL.
