Filtration (abstract 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 $\ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A$ of subspaces of $A$ such that

:$A=\bigcup_ F_$

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

:$\forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_.$

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'' $\mathcal(A)$ is defined as follows:
The multiplication is well-defined and endows $\mathcal(A)$ with the structure of a graded algebra, with gradation $\_.$ Furthermore if $A$ is associative then so is $\mathcal(A)$. Also if $A$ is unital, such that the unit lies in $F_0$, then $\mathcal(A)$ will be unital as well.

As algebras $A$ and $\mathcal(A)$ 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 $\bigoplus_^nG_i$ is isomorphic to $F_n$ as vector spaces).

Examples

Any graded algebra graded by $\mathbb$, for example $A = \bigoplus_ A_n$, has a filtration given by $F_n = \bigoplus_^n A_i$.

An example of a filtered algebra is the Clifford algebra $\operatorname(V,q)$ 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 $\mathfrak$ is also naturally filtered. The PBW theorem states that the associated graded algebra is simply $\mathrm (\mathfrak)$.

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.

See also

* Filtration (mathematics)
* Length function

References

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{{PlanetMath attribution, id=3938, title=Filtered algebra

Algebras
Homological algebra