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filtered algebra (Definition)
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
$\displaystyle \forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.$

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 graded algebra $ \mathcal{G}(A)$ is defined as follows:
  • As a vector space
    $\displaystyle \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, $
    where,
    $\displaystyle G_0=F_0,$   and $\displaystyle \forall n>0, \quad G_n=F_n/F_{n-1}\,,$
  • the multiplication is defined by
    $\displaystyle (x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}$
Theorem 3   The multiplication is well defined and endows $ \mathcal{G}(A)$ with the structure 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.



"filtered algebra" is owned by Dr_Absentius.
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Cross-references: isomorphic, exterior algebra, quadratic form, Clifford algebra, associative, well defined, vector space, graded algebra, multiplication, compatible, subspaces, filtration, algebra, field
There are 2 references to this entry.

This is version 8 of filtered algebra, born on 2003-01-28, modified 2007-02-04.
Object id is 3938, canonical name is FilteredAlgebra.
Accessed 2640 times total.

Classification:
AMS MSC: 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous)
Pending Errata and Addenda
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