# 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 $\left(A,cdot\right)$ over $k$ which has an increasing sequence $\left\{0\right\} subset F_0 subset F_1 subset cdots subset F_i subset cdots subset A$ of subspaces of $A$ such that

:$A=cup_\left\{iin mathbb\left\{N F_i$

and that is compatible with the multiplication in the following sense

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

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" $mathcal\left\{G\right\}\left(A\right)$ is defined as follows:

• As a vector space

:

where,

:$G_0=F_0,$ and

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

• the multiplication is defined by

:$\left(x+F_\left\{n\right\}\right)\left(y+F_\left\{m\right\}\right)=xcdot y+F_\left\{n+m+1\right\}$

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

As algebras $A$ and $mathcal\left\{G\right\}\left(A\right)$ 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 $mathrm\left\{Cliff\right\}\left(V,q\right)$ of a vector space $V$ endowed with a quadratic form $q.$ The associated graded algebra is , 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 $mathfrak\left\{g\right\}$ is also naturally filtered. The PBW theorem states that the associated graded algebra is simply $mathrm\left\{Sym\right\} \left(mathfrak\left\{g\right\}\right)$.

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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