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

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

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

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

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: Template:Unordered list The multiplication is well-defined and endows $𝒢(A)$ with the structure of a graded algebra, with gradation $\{G_n{\}}_{n\in \mathbb{N}}.$ Furthermore if $A$ is associative then so is $𝒢(A)$. 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 $\mathbb{N}$, for example $A={⨁}_{n\in \mathbb{N}}{A}_n$, 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 $𝔤$ is also naturally filtered. The PBW theorem states that the associated graded algebra is simply $\mathrm{S}\mathrm{y}\mathrm{m}(𝔤)$.

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 $T^{\ast }M$ which are polynomial along the fibers of the projection $\pi :T^{\ast }M\to M$.

The group algebra of a group with a length function is a filtered algebra.

See also

• Filtration (mathematics)
• Length function

References

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