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 k is an algebra (A,cdot) over k which has an increasing sequence {0} subset F_0 subset F_1 subset cdots subset F_i subset cdots subset A of subspaces of A such that

:A=cup_{iin mathbb{N F_i

and that is compatible with the multiplication in the following sense

: forall m,n in mathbb{N},qquad F_mcdot F_nsubset F_{n+m}.

Associated graded

In general there is the following construction that produces a graded algebra out of a filtered algebra.

If A as a filtered algebra then the "associated graded algebra" mathcal{G}(A) is defined as follows:

  • As a vector space

    : mathcal{G}(A)=igoplus_{nin mathbb{NG_n,,


    : G_0=F_0, and

    : forall n>0, quad G_n=F_n/F_{n-1},,

  • the multiplication is defined by

    : (x+F_{n})(y+F_{m})=xcdot y+F_{n+m+1}

The multiplication is well defined and endows mathcal{G}(A) with the structure of a graded algebra, with gradation {G_n}_{n in mathbb{N. Furthermore if A is associative then so is mathcal{G}(A).. Also if A is unital, such that the unit lies in F_0, then mathcal{G}(A). will be unital as well.

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


An example of a filtered algebra is the Clifford algebra mathrm{Cliff}(V,q) of a vector space V endowed with a quadratic form q. The associated graded algebra is igwedge 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 mathfrak{g} is also naturally filtered. The PBW theorem states that the associated graded algebra is simply mathrm{Sym} (mathfrak{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 is the commutative algebra of smooth functions on the cotangent bundle T^*M which are polynomial along the fibers of the projection pi:T^*M ightarrow M.


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