Filtered algebra explained

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.

k is an algebra (A, ⋅ ) over k that has an increasing sequence \{ 0 \} \subseteq F _{0} \subseteq F _{1} \subseteq … \subseteq F _{i} \subseteq … \subseteq A of subspaces of A such that A = c up _{i\in} F _{i}

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

\forall m,n \in N , F _{m ⋅} 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 l{G}(A) is defined as follows: The multiplication is well-defined and endows l{G}(A) with the structure of a graded algebra, with gradation \{G _{n\}} _{n} . Furthermore if A is associative then so is l{G}(A) . Also, if A is unital, such that the unit lies in F _{0} , then l{G}(A) will be unital as well.

As algebras

A and l{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 o plus ^{nG} _{i} is isomorphic to F _{n} as vector spaces).

Examples

Any graded algebra graded by

N , for example $A\; =\; \backslash bigoplus\_\; A\_n$, has a filtration given by . \operatorname{Cliff}(V,q) of a vector space V endowed with a quadratic form q. The associated graded algebra is w edge 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.

ak{g} is also naturally filtered. The PBW theorem states that the associated graded algebra is simply Sym ( ak{g}) .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 →} M .

The group algebra of a group with a length function is a filtered algebra.

See also

• Filtration (mathematics)
• Length function

References

• Book: Abe, Eiichi. Hopf Algebras. 1980. Cambridge University Press. Cambridge. 0-521-22240-0.