ИСТИНА

The purpose of this simple talk is to give few examples of questions and constructions, arising from a (rather naive) attempt to transfer some alge- braic structures on a Poisson manifold (M, π) to the quantized algebra of its functions. We begin with the case of a (finite-dimensional) Lie algebra action on M by Poisson vector fields; as one knows, every Poisson field gives rise (via the formality morphism) to a differentiation of the deformed algebra. However, this construction does not necessarily map commutators into commutators, the difference being given by an inner derivative. This gives a sequence of obstructions in a Chevalley cohomology of the Lie algebra with coefficients in Lichnerowicz-Poisson complex. Another interesting example, which I shall talk about are the so called Nijenhuis vector fields, which play an important role in the argument shift method and its generalizations. In this case we obtain obstructions in a twisted version of Hochschild cohomology.