ИСТИНА

Let (M, π) be a Poisson manifold, where π is a Poisson bivector, i.e. a section of the exterior square of the tangent bundle of M, whose Schouten bracket with itself vanishes. In this case one can introduce the Poisson bracket on functions by the formula: {f, g} = π(df, dg). Due to the well-known Kontsevich’s theorem, one can always find a formal deformation of the algebra of smooth functions on M, i.e. an associative ����-linear product ∗ in C∞(M)[[����]], the space of formal power series with coefficients in C∞(M), such that up to higher terms we shall have f ∗ g = fg + 1����{f, g} + ... 2 Moreover, this product is uniquely defined up to an equivalence relation. The algebra A(M) = (C∞)[[����]],∗) is often called the (formal) deformation quantization of M. One of important questions, related to this result, can be formulated in the following general form: what other algebraic and geometric structures (such as symmetries, integrable systems, complex or Kaehlerian structures, etc.) can be transferred to the deformation quantization of a manifold? In my talk I am going to address one of the simplest variant of this problem: suppose, there is a Lie algebra g acting on M by Poisson fields; is it possible to extend this action to the quantization A(M), i.e. to find a representation of g in the Lie algebra of differentiations on A(M)? It turns out, that this is the case, when dim g = 1, but in all other cases there are cohomological obstructions, which should vanish, if the answer is positive. In my talk I will explain the main ideas, that lie behind this result.