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 ${\displaystyle k}$ is an algebra ${\displaystyle (A,\cdot )}$ over ${\displaystyle k}$ that has an increasing sequence ${\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A}$ of subspaces of ${\displaystyle A}$ such that

${\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i))$

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

${\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}$

Associated graded algebra

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

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

The multiplication is well-defined and endows ${\displaystyle {\mathcal {G))(A)}$ with the structure of a graded algebra, with gradation ${\displaystyle \{G_{n}\}_{n\in \mathbb {N} }.}$ Furthermore if ${\displaystyle A}$ is associative then so is ${\displaystyle {\mathcal {G))(A)}$. Also if ${\displaystyle A}$ is unital, such that the unit lies in ${\displaystyle F_{0))$, then ${\displaystyle {\mathcal {G))(A)}$ will be unital as well.

As algebras ${\displaystyle A}$ and ${\displaystyle {\mathcal {G))(A)}$ are distinct (with the exception of the trivial case that ${\displaystyle A}$ is graded) but as vector spaces they are isomorphic. (One can prove by induction that ${\displaystyle \bigoplus _{i=0}^{n}G_{i))$ is isomorphic to ${\displaystyle F_{n))$ as vector spaces).

Examples

Any graded algebra graded by ${\displaystyle \mathbb {N} }$, for example ${\textstyle A=\bigoplus _{n\in \mathbb {N} }A_{n))$, has a filtration given by ${\textstyle F_{n}=\bigoplus _{i=0}^{n}A_{i))$.

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

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

Scalar differential operators on a manifold ${\displaystyle M}$ 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 ${\displaystyle T^{*}M}$ which are polynomial along the fibers of the projection ${\displaystyle \pi \colon T^{*}M\rightarrow M}$.

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