(Bi)modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions. (English) Zbl 1099.53061

Molitor-Braun, Carine (ed.) et al., Proceedings of the 4th conference on Poisson geometry, Luxembourg, June 7–11, 2004. Luxembourg: Université du Luxembourg (ISBN 2-87971-253-X/pbk). Travaux Mathématiques 16, 9-40 (2005).
Some algebraic questions in the deformation quantization are presented with the main emphasize on the following aspects: (i) the algebra morphisms, (ii) the modules and (iii) the commutants of modules. It is shown that to each of these topics there is a corresponding geometric situation in “the classical limit”: (i) the Poisson maps, (ii) the coistropic maps, and (iii) the phase space reduction. It is argued that the quantization problem in these three cases is very important and it finds many applications in quantum physics such as quantization of symmetries and integrable systems, quantization of first class constraints and quantization of the reduced Poisson algebra. The star-product on the associative noncommutative algebras is defined and its reduction is studied. Open questions with respect to the quantization of Poisson maps (symplectic-to symplectic case), the quantization of coisotropic submanifolds and the quantization of phase space reduction are also given.
For the entire collection see [Zbl 1085.53003].


53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods