Deformation quantization of integrable systems

TL;DR

This paper investigates conditions under which a Poisson algebra can be deformed so that a Poisson-commutative subalgebra remains commutative, introducing cohomological obstructions related to Hochschild and Poisson cohomology.

Contribution

It introduces a new framework of cohomological obstructions for deformation quantization preserving commutativity of subalgebras, generalizing previous results for polynomial cases.

Findings

Defined cohomological obstructions in Hochschild cohomology

Reduced obstructions to Poisson relative cohomology in specific cases

Connected new obstructions with previously known classes by Garay and van Straten

Abstract

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