ag.algebraic geometry - Examples of Poisson Schemes

A Poisson Manifold is a real manifold $M$ along with a Lie bracket $[cdot,cdot]$ on $C^infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among them:

1. You can define the notion of an integrable system structure on a Poisson manifold, which allows them to be applied to solving problems in physics with enough symmetry.


2. Every symplectic manifold is automatically Poisson


3. Any Poisson manifold has a foliation by symplectic leaves.

(Reference for all of this: anything on Poisson manifolds, in particular, wikipedia.)

Now, I've seen people seriously (for instance, in Vanhaecke's book) extend this notion to affine varieties over $mathbb{C}$, where being Poisson means that the structure sheaf is a sheaf of Poisson Algebras, specifically, a Poisson algebra is an associative algebra along with a Lie bracket that is a derivation in each variable.

Now, what I'm interested in is how far this can be generalized and still have something where there are interesting (new!) examples. For instance, is "Poisson Scheme $X$ over $S$" a real object of interest? Specifically, I'm wondering if there are any examples where $S$ is not the spectrum of a field of characteristic zero, say $S$ is a finite field, or something positive dimensional, or nonreduced, etc, and if there are examples of this form, what makes them interesting? For instance, one reason that Poisson manifolds are interesting is that they are applicable to physics and, in fact in many cases to problems related to the geometry of the moduli space of vector bundles on a Riemann surface.