Filtered algebra

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 formula_1 is an algebra formula_2 over formula_1 that has an increasing sequence formula_4 of subspaces of formula_5 such that 
and that is compatible with the multiplication in the following sense: 
Associated graded algebra.
In general there is the following construction that produces a graded algebra out of a filtered algebra. 
If formula_5 is a filtered algebra then the "associated graded algebra" formula_9 is defined as follows: G_n\, </math>
where, 
for all formula_13 and formula_14. (More precisely, the multiplication map formula_15 is combined from the maps
for all formula_17 and formula_18.) 
The multiplication is well defined and endows formula_9 with the structure of a graded algebra, with gradation formula_20 Furthermore if formula_5 is associative then so is formula_9. Also if formula_5 is unital, such that the unit lies in formula_24, then formula_9 will be unital as well.
As algebras formula_5 and formula_9 are distinct (with the exception of the trivial case that formula_5 is graded) but as vector spaces they are isomorphic.
Examples.
Any graded algebra graded by ℕ, for example formula_29, has a filtration given by formula_30. 
An example of a filtered algebra is the Clifford algebra formula_31 of a vector space formula_32 endowed with a quadratic form formula_33 The associated graded algebra is formula_34, the exterior algebra of formula_35 
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 formula_36 is also naturally filtered. The PBW theorem states that the associated graded algebra is simply formula_37.
Scalar differential operators on a manifold formula_38 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 formula_39 which are polynomial along the fibers of the projection formula_40.
The group algebra of a group with a length function is a filtered algebra.