Filtered Algebra - Examples

Examples

Any graded algebra graded by ℕ, 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 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.