Deformation quantization of pseudo-symplectic (Poisson) groupoids

We introduce a new kind of groupoid—a pseudo-étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson algebras in the framework of deformation quantization. Received: September 2004 Revision: September 2005 Accepted: September 2005 Dedicated to A. Weinstein on his 60th birthday