filtered algebra

Definition 1.

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

$$\forall m,n\in \mathbb{N},F_m\cdot F_n\subset F_{n+m}.$$

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

Definition 2.

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

• As a vector space

  $$𝒢(A)=\underset{n\in \mathbb{N}}{\oplus }{G}_n,$$

  where,

  $$G_0=F_0,\text{and }\forall n>0,G_n=F_n/F_{n-1},$$

• 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 $\mathrm{G}\mathit{}\mathrm{(}A\mathrm{)}$ with the of a graded algebra, with gradation ${\mathrm{\{}{G}_n\mathrm{\}}}_{n\mathrm{\in }\mathrm{N}}$. Furthermore if $A$ is associative then so is $\mathrm{G}\mathit{}\mathrm{(}A\mathrm{)}$.

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 $V$.

As algebras $A$ and $𝒢(A)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic.

Theorem 4.

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

Title	filtered algebra
Canonical name	FilteredAlgebra
Date of creation	2013-03-22 13:23:55
Last modified on	2013-03-22 13:23:55
Owner	Dr_Absentius (537)
Last modified by	Dr_Absentius (537)
Numerical id	11
Author	Dr_Absentius (537)
Entry type	Definition
Classification	msc 08A99