How to compute the deformation quantizations of a polynomial Poisson algebra?

As a specialist told me, given a smooth affine Poisson algebra $S$ over $\mathbb{C}$, up to a choice of certain characteristic class, one can find one of the deformation quantizations of $S$, say $A$, such that the Poisson cohomology of $S$, suitably tensored with $\mathbb{C}[[h]]$, is isomorphic to the Hochschild cohomology of $A$. (Maybe my memory is not very good, the above is what I remember now. )

My question is, given $S$, how to compute explicitely $A$.

For example, if $S=\mathbb{C}[X, Y]$ with $\{X, Y\}=XY$, how to compute the "right" deformation quantization? I guess that $A=\mathbb{C}[[h]]<X, Y>/(XY-(h+1)YX)$

What about polynomial Poisson algebras in general?

• Your algebra A does not specialize to the correct one when h=0, so we should at least guess the relation is xy-g(h)yx with g(0)=1. On the other hand your problem is very difficult even in the example you ask about--- See the top of page 8 of ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf – Daniel Pomerleano Mar 1 '17 at 16:02

Not a complete answer but some observations. First it might be necessary not to take formal power series but formal Laurent series in order to get a reasonable behaviour of the Hochschild cohomology. The reason is that say in the symplectic case where the relation would be $[X, Y] = \hbar 1$ you would like all derivations to be inner (first Hochschild cohomology zero). What you can prove is that the derivations are "quasi-inner" in the sense that every derivation is of the form $D = \frac{1}{\hbar} \mathrm{ad}(a)$ with some algebra element $a$. So if working over the formal power series, such derivations are "outer" since the hypothetical element $\frac{a}{\hbar}$ is not part of your algebra. Then the Hochschild cohomology has weird contributions in zeroth order of $\hbar$ and is trivial in higher order (you can check this rather easily, say for the first Hochschild cohomology)