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:


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

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

If $$A$$ is a filtered algebra, then the associated graded algebra $$\mathcal{G}(A)$$ is defined as follows: 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 $$\mathcal{G}(A)$$. Also, if $$A$$ is unital, such that the unit lies in $$F_0$$, then $$\mathcal{G}(A)$$ will be unital as well.

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. (One can prove by induction that $$\bigoplus_{i=0}^nG_i$$ is isomorphic to $$F_n$$ as vector spaces).

Examples
Any graded algebra graded by $$\mathbb{N}$$, for example $A = \bigoplus_{n\in \mathbb{N}} A_n $, has a filtration given by.

An example of a filtered algebra is the Clifford algebra $$\operatorname{Cliff}(V,q)$$ 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 $$\mathrm{Sym} (\mathfrak{g})$$.

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 $$\pi\colon T^*M\rightarrow M$$.

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