TITLE: What are important examples of filtered/graded rings in physics?
QUESTION [11 upvotes]: Hi,
what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics? 
Of course also mathematicians may answer from their perspective :)
Edit: Information on my background: My "primeval" motivation to understand filtered rings/graded rings, comes from mathematics: They are the basics for D-Module theory. 
Though I would also like to see these rings from a different perspective, therefor I asked the question.
 My knowledge in physics is a bit limited, I only attended some undergraduate courses (quantum mechanics, electro dynamics, classical mechanics). Though more sophisticated answers are fine, I don't need to understand every detail, I just want to get the flavor.

REPLY [4 votes]: The deformation quantization construction, famously carried into full generality by Kontsevich, is the construction, from the datum of a manifold $M$ and a Poisson structure $\{\mathord-,\mathord-\}$ on it, of a generally non-commutative associative algebra $\mathcal A$ which is endowed with a filtration such that the associated graded algebra (which is naturally a Poisson algebra) $\mathrm{gr}\mathcal A$ is isomorphic to the algebra of smooth functions on $M$ and its Poisson structure. This should be an example of interest to physicists! (Also, this includes José's example 1, as the Weyl algebra is a deformation quantization of the ring of (polynomial) functions on the plane, and more or less example 2)
In general, a filtered object $X$ is an objects with a chosen aproximation, given by its associated graded object $\mathrm{gr}\\,X$, which ideally reflects some of the interesing propoerties of $X$ but which is, at the same time, simpler than $X$. Indeed, one can in most cases think of elements of $\mathrm{gr}\\,X$ as «elements of $X$ up to details».