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 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: 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: 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). 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.