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,,

  where,

  :$$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.

Examples

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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