mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebraand representation theory.
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.
If as a filtered algebra then the "associated graded algebra" is defined as follows:
- As a vector space
- the multiplication is defined by
The multiplication is well defined and endows with the structure of a graded algebra, with gradation Furthermore if is
associativethen so is . Also if is unital, such that the unit lies in , then will be unital as well.
As algebras and are distinct (with the exception of the trivial case that is graded) but as vector spaces they are isomorphic.
symmetric algebraon the dual of an affine spaceis a filtered algebra of polynomials; on a vector space, one instead obtains a graded algebra.
universal enveloping algebraof a Lie algebrais also naturally filtered. The PBW theoremstates 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 is the commutative algebra of smooth functions on the cotangent bundle which are polynomial along the fibers of the projection .
- As a vector space
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