Deformation Quantization of Momentum Maps and Integrable Systems

Deformation Quantization according to Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer (1978) consists in a formal associative deformation (a so-called star-product) of the commutative algebra $A$ of all smooth complex-valued functions on a Poisson manifold such that the formal parameter corresponds to Planck's constant and the first order commutator is proportional to the Poisson bracket. Existence and classification of star-products have been shown by DeWilde-Lecomte (1983, symplectic case), Nest-Tsygan (1995, symplectic case) and M.Kontsevitch (1997, general Poisson case). We show how to quantize a morphism of a finite-dimensional Lie algebra in $A$ (a so-called momentum map) in the case of symplectic manifolds. A Liouville-integrable Hamiltonian system is a particular case of this for an abelian Lie algebra.