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 (A,\cdot) over k that has an increasing sequence of subspaces of A 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 A is a filtered algebra, then the associated graded algebra is defined as follows: The multiplication is well-defined and endows with the structure of a graded algebra, with gradation Furthermore if A is associative then so is. Also, if A is unital, such that the unit lies in F_0, then will be unital as well. As algebras A and are distinct (with the exception of the trivial case that A is graded) but as vector spaces they are isomorphic. (One can prove by induction that is isomorphic to F_n as vector spaces).

Examples

Any graded algebra graded by \mathbb{N}, for example, has a filtration given by. An example of a filtered algebra is the Clifford algebra of a vector space V endowed with a quadratic form q. The associated graded algebra is \bigwedge V, the exterior algebra of V. 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 M 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 T^*M which are polynomial along the fibers of the projection. The group algebra of a group with a length function is a filtered algebra.