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

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

and that is compatible with the multiplication in the following sense

\forall m,n\in {\mathbb {N}},\qquad F_{m}\cdot F_{n}\subset 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

$\forall n>0,\quad G_{n}=F_{n}/F_{{n-1}}\,,$
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.

Examples

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

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 ${\mathfrak {g}}$ 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.

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Filtered algebra

Associated graded algebra

Examples

See also

References