# 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 in the 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 is the following construction that produces a graded algebra out of a filtered algebra.

###### 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 associated graded algebra is $\bigwedge V$, the exterior algebra of $V$.

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.

###### Theorem 4.

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

Title filtered algebra FilteredAlgebra 2013-03-22 13:23:55 2013-03-22 13:23:55 Dr_Absentius (537) Dr_Absentius (537) 11 Dr_Absentius (537) Definition msc 08A99