filtered algebra

Definition 1.

A filtered algebra over the field $k$ is an algebra $(A,\\cdot)$ over $k$ which is endowed with a filtration $\\mathcal{F}=\\{F_{i}\\}_{i\\in\\mathbb{N}}$ by subspaces, compatible with the multiplication inthe following sense

\\forall m,n\\in\\mathbb{N},\\qquad F_{m}\\cdot F_{n}\\subset F_{n+m}.

A special case of filtered algebra is a graded algebra. In general there isthe following construction that produces a graded algebra out of a filteredalgebra.

Definition 2.

Let $(A,\\cdot,\\mathcal{F})$ be a filtered algebra then the associated http://planetmath.org/node/3071graded algebra $\\mathcal{G}(A)$ is defined as follows:

•
As a vector space
$\\mathcal{G}(A)=\\bigoplus_{n\\in\\mathbb{N}}G_{n}\\,,$
where,
$G_{0}=F_{0},\\quad\\text{and }\\forall n>0,\\quad G_{n}=F_{n}/F_{n-1}\\,,$
•
the multiplication is defined by
$(x+F_{n})(y+F_{m})=x\\cdot y+F_{n+m}$

Theorem 3.

The multiplication is well defined and endows $\\mathcal{G}(A)$ with the of a graded algebra, with gradation $\\{G_{n}\\}_{n\\in\\mathbb{N}}$ .Furthermore if $A$ is associative then so is $\\mathcal{G}(A)$ .

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 associatedgraded algebra is $\\bigwedge V$ , the exterior algebra of $V$ .

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

Theorem 4.

The underlying vector spaces of $A$ and $\\mathcal{G}(A)$ are isomorphic.