# Filtered algebra

An algebra $S$ in which there are distinguished subspaces $S _ \alpha$, indexed by the elements of a totally ordered group $A$( most often $A$ is the additive group of integers $\mathbf Z$), such that $S _ \alpha \subseteq S _ \beta$ for $\alpha < \beta$ and $S _ \alpha S _ \beta \subseteq S _ {\alpha + \beta }$( an increasing filtration). Sometimes one considers the case where $S _ \alpha \supseteq S _ \beta$ for $\alpha < \beta$( a decreasing filtration), but it reduces to the preceding case by reversing the order in $A$. With each filtered algebra $S$ one associates the graded algebra

$$\mathop{\rm gr} S = \ \oplus _ \alpha \overline{S}\; _ \alpha ,$$

where $\overline{S}\; _ \alpha = S _ \alpha / \sum _ {\beta < \alpha } S _ \beta$( if $A = \mathbf Z$, then $\overline{S}\; _ \alpha = S _ \alpha /S _ {\alpha - 1 }$), and the product of the elements $\overline{x}\; \in \overline{S}\; _ \alpha$ and $\overline{y}\; \in \overline{S}\; _ \beta$ is defined by the formula $\overline{x}\; \overline{y}\; = \overline{xy}\;$, where $x$ and $y$ are representatives of the cosets $\overline{x}\;$ and $\overline{y}\;$, and $\overline{xy}\;$ is the coset of $\sum _ {\gamma < \alpha + \beta } S _ \gamma$ generated by $xy \in S _ {\alpha + \beta }$. If some multilinear identity is satisfied in $S$( for example, commutativity, associativity, or the Jacobi identity), then it is also satisfied in $\mathop{\rm gr} S$.

### Examples.

1) Let $S$ be a Clifford algebra and let $S _ {n}$, $n \in \mathbf Z$, be the collection of elements that can be represented in the form of (non-commutative) polynomials of degree $\leq n$ in the generators. One obtains an increasing $\mathbf Z$- filtration of $S$ in which $S _ {n} = 0$ for $n < 0$. The associated graded algebra is the exterior algebra with the same number of generators.

2) In the universal enveloping algebra of a Lie algebra one can define an increasing $\mathbf Z$- filtration in the same way as in the preceding example. By the Birkhoff–Witt theorem, the associated graded algebra is the polynomial algebra.

How to Cite This Entry:
Filtered algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Filtered_algebra&oldid=46920
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article