Symplectic double groupoids over Poisson $(ax+b)$-groups

Symplectic Groupoids and Poisson Manifolds

0. Introduction. A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure. The identity elements in T turn out to form a Poisson manifold To? and the correspondence between symplectic groupoids and Poisson manifolds is a natural extension of the one between Lie groups and Lie algebras. As with Lie groups, und...

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 manifolds in the framework of deformation quantizatio...

Poisson Sigma Models and Symplectic Groupoids

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Pois...

Poisson Fibrations and Fibered Symplectic Groupoids

We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.

Lie, symplectic and Poisson groupoids and their Lie algebroids