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

$A=\cup_{i\in \mathbb{N}} F_i$

and that is compatible with the multiplication in the following sense

$\forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n%2Bm}.$

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

As a vector space

$\mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,,$

where,

$G_0=F_0,$ and

$\forall n>0, \quad G_n=F_n/F_{n-1}\,,$
the multiplication is defined by

$(x%2BF_{n-1})(y%2BF_{m-1})=x\cdot y%2BF_{n%2Bm-1}$

for all $x\in F_n$ and $y\in F_m$ . (More precisely, the multiplication map $\mathcal{G}(A)\times \mathcal{G}(A) \to \mathcal{G}(A)$ is combined from the maps

$(F_n / F_{n-1}) \times (F_m / F_{m-1}) \to F_{n%2Bm}/F_{n%2Bm-1}, \ \ \ \ \ \left(x%2BF_{n-1},y%2BF_{m-1}\right) \mapsto x\cdot y%2BF_{n%2Bm-1}$

for all $n\geq 0$ and $m\geq 0$ .)

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

Any graded algebra graded by ℕ, for example $A = \oplus_{n\in \mathbb{N}} A_n$ , has a filtration given by $F_n = \oplus_{i=0}^n A_i$ .

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 $\bigwedge V$ , 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 $\pi\colon T^*M\rightarrow M$ .

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

Filtered algebra

Associated graded algebra

Examples

See also