ИСТИНА

Let M,π be a Poisson manifold; according to Kontsevich’s theorem, one can always find a deformed noncommutative product on the space of formal power series in 􏰛 with coefficients in C∞(M), such that for any two f,g ∈ C∞(M), their commutator with respect to the new product will coincide with the Poisson bracket up to 􏰛2 terms. This algebra is usually called deformation quantization of M. In my talk I shall address the question, whether one can extend a given commutative (with respect to the Poisson bracket) family of functions on M to a commutative family of elements in its deformation quantization and its straightforward generalization to the case of a Lie algebra action on this manifold. Due to Darboux theorem this question can always be solved locally, when the manifold M is symplectic and the commutative functions are independent, thus the question is closely related to the global properties of the singular sets of the commutative system. In my talk I shall describe various cohomological obstructions for this procedure and discuss possible relations between them.