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:

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

where,

$G_{0}=F_{0},$ and

$\forall n>0,\ G_{n}=F_{n}/F_{n-1}\,,$
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 ${\mathcal {G}}(A)\times {\mathcal {G}}(A)\to {\mathcal {G}}(A)$ is combined from the maps

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

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 $\bigoplus _{i=0}^{n}G_{i}$ is isomorphic to $F_{n}$ as vector spaces).

Examples

Any graded algebra graded by $\mathbb {N}$ , 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 $\operatorname {Cliff} (V,q)$ 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.

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.