Filtered algebra

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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 ${\displaystyle k}$ which has an increasing sequence ${\displaystyle \{0\}\subset F_{0}\subset F_{1}\subset \cdots \subset F_{i}\subset \cdots \subset A}$ of subspaces of ${\displaystyle A}$ such that

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

and that is compatible with the multiplication in the following sense

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

Contents

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:

• As a vector space
${\displaystyle {\mathcal {G}}(A)=\bigoplus _{n\in \mathbb {N} }G_{n}\,,}$

where,

${\displaystyle G_{0}=F_{0},}$ and
${\displaystyle \forall n>0,\quad G_{n}=F_{n}/F_{n-1}\,,}$
• the multiplication is defined by
${\displaystyle (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}}$

for all ${\displaystyle x\in F_{n}}$ and ${\displaystyle y\in F_{m}}$. (More precisely, the multiplication map ${\displaystyle {\mathcal {G}}(A)\times {\mathcal {G}}(A)\to {\mathcal {G}}(A)}$ is combined from the maps

${\displaystyle (F_{n}/F_{n-1})\times (F_{m}/F_{m-1})\to F_{n+m}/F_{n+m-1},\ \ \ \ \ \left(x+F_{n-1},y+F_{m-1}\right)\mapsto x\cdot y+F_{n+m-1}}$
for all ${\displaystyle n\geq 0}$ and ${\displaystyle m\geq 0}$.)

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.

Examples

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

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