Homological obstructions to quantization commutative families of functions and actions of Lie algebras

I will tell on which obstructions exist for solving the problem of prolongation of a family of commuting (with respect to the Poisson bracket) functions on a Poisson manifold to the family of commuting elements in the deformation quantization of this manifold. This problem is closely related to the question of prolongation of an action of a Lie algebra on this manifold to an action of the same algebra on deformed (quantized) algebra of its functions, on which I also will tell, as well as on relation of these two problems to each other. At the end I will speculate on to what these cohomological obstacles turn to in the case of symplectic manifold.