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.