Filtered algebra

From Wikipedia, the free encyclopedia

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.

A filtered algebra over the field $k$ is an algebra $(A,\cdot)$ over $k$ which is endowed with a filtration

$\mathcal{F}=\{F_i\}_{i\in \mathbb{N}}$

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.

[edit] Associated graded

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

If $(A,\cdot,\mathcal{F})$ as a filtered algebra then the associated graded algebra $\mathcal{G}(A)$ is defined as follows:

As a vector space

$\mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,,$

where,

$G 0 = F 0,$ and

$\forall n>0, \quad G_n=F_n/F_{n-1}\,,$
the multiplication is defined by

$(x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}$

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 $\mathcal{G}(A).$

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.

[edit] Examples

An example of a filtered algebra is the Clifford algebra $Cliff(V, q)$ of a vector space $V$ endowed with a quadratic form $q .$ The associated graded algebra is $\bigwedge V$ , the exterior algebra of $V .$

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is also naturally filtered. The PBW theorem states that the associated graded algebra is simply $\mathrm{Sym} (\mathfrak{g})$ .