Filtered algebra

From Wikipedia, the free encyclopedia

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 is endowed with a filtration

\mathcal{F}=\{F_i\}_{i\in \mathbb{N}}

compatible with the multiplication in the following sense

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

A special case of filtered algebra is a graded algebra.

[edit] Associated graded

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

If (A,\cdot,\mathcal{F}) as 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,
    G0 = F0, and
    \forall n>0, \quad G_n=F_n/F_{n-1}\,,
  • the multiplication is defined by
    (x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}

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).

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.

[edit] Examples

An example of a filtered algebra is the Clifford algebra 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 is the commutative algebra of smooth functions on the cotangent bundle T * M which are polynomial along the fibers of the projection \pi:T^*M\rightarrow M.

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the GFDL.