Filtered algebra

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 G ( A ) is defined as follows:

The multiplication is well defined and endows G ( A ) with the structure of a graded algebra, with gradation { G n } n ∈ N . Furthermore if A is associative then so is G ( A ) . Also if A is unital, such that the unit lies in F 0 , then G ( A ) will be unital as well.

As algebras A and G ( A ) are distinct (with the exception of the trivial case that A is graded) but as vector spaces they are isomorphic.

Examples

Any graded algebra graded by ℕ, for example A = ⊕ n ∈ N A n , has a filtration given by F n = ⊕ i = 0 n A i .

An example of a filtered algebra is the Clifford algebra C l i f f ( V , q ) of a vector space V endowed with a quadratic form q . The associated graded algebra is ⋀ 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 g is also naturally filtered. The PBW theorem states that the associated graded algebra is simply S y m ( g ) .

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 π : T ∗ M → M .

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