ИСТИНА

As one knows, every Poisson manifold M admits a deformation quantization, i.e. an associative *-product on the space of formal power series in a variable \hbar with coefficients in the algebra of smooth functions on M, coherent with the Poisson bracket on M. In my talk I will address the question, whether for a Poisson commutative subalgebra S in C^\infty(M) there exists an extension of S to a commutative subalgebra with respect to the *-product. This question (sometimes referred to as the quantum integrability) is closely related with the problem of finding an extension of a Lie algebra action on M to its action (by derivatives) on the deformed algebra. Both questions are in a great measure still open and depend on the structure of the singularities of the corresponding foliation. In my talk I will try to describe the few known facts about these problems, and give some concrete examples of these constructions.