Computations of certain Poisson cohomology groups I am reading the paper  Grothendieck groups of Poisson vector bundles by Viktor L. Ginzburg.
In that paper, the author introduces a new invariant for Poisson manifolds; which is called as the Poisson K-ring.
One justification for introduction of this new invariant is that (as of 2001), the existing invariants related to Poisson manifolds are difficult to compute. For example, "Poisson cohomology and its derivatives are not calculated explicitly even for dual spaces to semisimple Lie algebra".
Question : What is the current status regarding computation of Poisson cohomology groups of semisimple Lie algebras? Are there any techniques introduced after 2001 that have made it easier to compute?
Side question: It is not clear what "its derivatives" means in the above quoted statement. Can some one clarify what this means?
 A: The Poisson cohomology associated to compact Lie algebras was computed by Ginzburg and Weinstein already in 1992:
Lie Poisson structure on some Poisson Lie groups, Thm. 3.5
For the non-compact semisimple case Weinstein constructed some non-trivial cohomology classes in
Poisson geometry of the principal series and nonlinearizable structures
to prove non-linearizablity for most cases.
In my thesis I managed to compute the Poisson cohomology of $sl_2(\mathbb{R})$ and $so(3,1)$.
The Poisson cohomology and linearization of $sl_2(\mathbb{R})$ and $sl_2(\mathbb{C})$
or also:
The Poisson cohomology of $sl_2(\mathbb{R})$
Some techniques therein are quite general, but some are very specific for the corresponding Lie algebras. We view Poisson cohomology as Lie algebra with coefficients. The main idea is to divide the calculation into a "formal part" at the singularities and a "flat part". In these two cases that basically means that one considers the short exact sequence
$0\to \mathrm{ker} j^\infty_0 \to C^\infty(\mathfrak{g}^*)\xrightarrow{j^\infty_0} \mathbb{R}[[\mathfrak{g}]] \to 0$
which induces a long exact sequence in cohomology, and it turns out that one can compute the cohomology groups associated to the left and right term. The point here is that the Poisson structures for both Lie algebras have singularities only at the origin. In general for a Poisson structure with more singularities one could try to adopt this method for example by dealing with the different strata stepwise.
PS: In fact  one of my current research questions is precisely how/if one can use that techniques to also compute the Poisson cohomology groups associated to other semisimple Lie algebras. If you are interested in the topic/some discussion feel free to contact me.
