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

  :

  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 , 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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