ИСТИНА

As one knows, for every Poisson manifold M there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra g act by derivations on the functions on M. The question, which we shall address in my talk is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of g is 1, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of g is greater than 1, this method does not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension. We hope that these obstructions can be of interest in their own right as invariants of Poisson actions. The talk is based on the paper arXiv:1612.02673.