ИСТИНА

Let A be a Poisson algebra (i.e. a commutative algebra with antisymmetric bracket, that verifies Leibnitz rule with respect to the product and the Jacobi identity). As it is known, Kontsevich’s theorem provides a deformation quantization of this algebra, i.e. a series of bidifferential operators ����k>0 ����Bk, such that f ∗ g = fg + ���� ����\sum Bk(f, g) h^k determines an associative ����-linear product in A[[h����]] with commutator, equal to the Poisson bracket in A up to the higher degree terms in ����. Suppose now, that there is a Lie algebra g acting on A by derivatives. The question is, if it is always possible to extend this action to the deformed algebra A[[h����]]. In my talk I will define a series of cohomological obstructions to this. Then I will discuss the application of this theory to the case, when g is a commutative Lie algebra of Hamiltonian vector fields on a Poisson manifold M, A = C∞(M), for instance, to the algebra, determined by the argument shift method and its generalizations, associated to a bihamiltonian structure.