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 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.

is a filtered algebra, then the associated graded algebra is defined as follows: is associative is unital will be unital as well.

is graded) but as vector spaces they are isomorphic. (One can prove by induction as vector spaces).

Examples

Any graded algebra.

An example of a filtered algebra is the Clifford algebra endowed with a quadratic form, the exterior algebra

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.

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.

The group algebra of a group with a length function is a filtered algebra.

See also

• Filtration (mathematics)

• Length function

References