08A99-FilteredAlgebra.tex

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\pmtitle{filtered algebra}
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\begin{document}
\begin{defn}
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
$$\forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.$$
\end{defn}

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.

\begin{defn}
Let $(A,\cdot,\mathcal{F})$ be a filtered algebra then the \emph{associated \PMlinkid{graded algebra}{3071}} $ \mathcal{G}(A)$ is defined as follows:
\begin{itemize}
\item As a vector space  
$$ \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, $$
where,
$$G_0=F_0,\quad \text{and } \forall n>0, \quad G_n=F_n/F_{n-1}\,,$$
\item the multiplication is defined by
$$(x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}$$   
\end{itemize}
\end{defn}

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

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.  

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