Filtered algebra

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 $k$ that has an increasing sequence $\{0\}\subseteq F_0\subseteq F_1\subseteq \cdots \subseteq F_i\subseteq \cdots \subseteq A$ of subspaces of $A$ such that

$A=\bigcup _{i\in \mathbb{\mathbb{N}}}{F}_i$

and that is compatible with the multiplication in the following sense:

$\mathrm{\forall }m,n\in \mathbb{\mathbb{N}},F_m\cdot F_n\subseteq F_{n+m}.$

Associated graded algebra

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

If

$A$

is a filtered algebra, then the associated graded algebra

${𝒢}(A)$

is defined as follows:

• As a vector space

  ${𝒢}(A)=\underset{n\in \mathbb{\mathbb{N}}}{⨁}{G}_n,$

  where,

  $G_0=F_0,$ and

  $$\begin{gathered} \mathrm{\forall }n>0, \\ {G}_n=F_n/F_{n-1}, \end{gathered}$$

• the multiplication is defined by
  $(x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}$

  for all $x\in F_n$ and $y\in F_m$. (More precisely, the multiplication map ${𝒢}(A)\times {𝒢}(A)\to {𝒢}(A)$ is combined from the maps

  $$\begin{gathered} (F_n/F_{n-1})\times (F_m/F_{m-1})\to F_{n+m}/F_{n+m-1}, \\ (x+F_{n-1},y+F_{m-1})\mapsto x\cdot y+F_{n+m-1} \end{gathered}$$

  for all $n\ge 0$ and $m\ge 0$.)

The multiplication is well-defined and endows ${𝒢}(A)$ with the structure of a graded algebra, with gradation $\{G_n{\}}_{n\in \mathbb{\mathbb{N}}}.$ Furthermore if $A$ is associative then so is ${𝒢}(A)$. Also, if $A$ is unital, such that the unit lies in $F_0$, then ${𝒢}(A)$ will be unital as well. As algebras $A$ and ${𝒢}(A)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic. (One can prove by induction that $\underset{i=0}{\overset{n}{⨁}}{G}_i$ is isomorphic to $F_n$ as vector spaces).

Examples

Any graded algebra graded by $\mathbb{\mathbb{N}}$, for example $A=\underset{n\in \mathbb{\mathbb{N}}}{⨁}{A}_n$, has a filtration given by $F_n={⨁}_{i=0}^n{A}_i$. 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 $V.$ The symmetric algebra on the dual of an affine space is a filtered algebra of polynomials; on a vector space, one instead obtains a graded algebra. 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 $\mathrm{S}\mathrm{y}\mathrm{m}(\mathfrak{𝔤})$. Scalar differential operators on a manifold $M$ form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded algebra is the commutative algebra of smooth functions on the cotangent bundle $T^{*}M$ which are polynomial along the fibers of the projection $\pi :T^{*}M\to M$. The group algebra of a group with a length function is a filtered algebra.

See also

• Filtration (mathematics)
• Length function

References

• Abe, Eiichi (1980). Hopf Algebras. Cambridge: Cambridge University Press. ISBN 0-521-22240-0.

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.