and that is compatible with the multiplication in the following sense
Associated graded algebra
In general there is the following construction that produces a graded algebra out of a filtered algebra.
If is a filtered algebra then the associated graded algebra is defined as follows:
- As a vector space
- the multiplication is defined by
for all and . (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 Furthermore if is associative then 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.
Any graded algebra graded by ℕ, for example , has a filtration given by .
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 .