Deformation quantization of integrable systems

Abstract

In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra $\mathcal{A}$ so that certain Poisson-commutative subalgebra $\mathcal{C}$ in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of $\mathcal{C}$ with coefficients in $\mathcal{A}$. In some particular case of the pair $(\mathcal{A},\mathcal{C})$ we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra $\mathcal{C}$ is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.