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
that has an increasing sequence
such that and that is compatible with the multiplication in the following sense: In general, there is the following construction that produces a graded algebra out of a filtered algebra.
is a filtered algebra, then the associated graded algebra
(More precisely, the multiplication map
is combined from the maps The multiplication is well-defined and endows
with the structure of a graded algebra, with gradation
is unital, such that the unit lies in
are distinct (with the exception of the trivial case that
is graded) but as vector spaces they are isomorphic.
(One can prove by induction that
as vector spaces).
Any graded algebra graded by
An example of a filtered algebra is the Clifford algebra
of a vector space
endowed with a quadratic form
The associated graded algebra is
, the exterior algebra of
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
is also naturally filtered.
The PBW theorem states that the associated graded algebra is simply
Scalar differential operators on a manifold
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
which are polynomial along the fibers of the projection
The group algebra of a group with a length function is a filtered algebra.
This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.