Morita Equivalent Symplectic Groupoids

Abstract

Morita equivalence of C*-algebras, first introduced by M. Rieffel, has been widely accepted as one of the most important equivalence relations in C*-algebras [Riel] [Rie2] [Rie3] [Rie4]. Roughly speaking, two C*- algebras are said to be Morita equivalent if there is an equivalence bimodule between them. Morita equivalent C*-algebras have many similar features. For instance, they have equivalent categories of left modules, isomorphic K-groups, and so on. Also, Morita equivalence plays a very important role in understanding the structure of some C*-algebras such as transformation C*-algebras and foliation C*-algebras. A natural question arises as to what the classical analogue of this equivalence relation is. It is generally accepted that the classical analogue of a C*-algebra (or non-commutative algebra) is a Poisson manifold. So, more precisely, we expect to find an equivalence relation for Poisson manifolds that plays the same role as Morita equivalence does for C*-algebras. A solution to this problem was made possible by the recent introduction of symplectic groupoids in the study of Poisson manifolds due to Karasev and Weinstein [CDW] [Ka] [W2]. The original purpose for introducing symplectic groupoids was to study nonlinear commutation relations and quantization theory. In fact, it turns out that symplectic groupoids provide a bridge between Poisson manifolds, C*-algebras, as well as quantizations. Therefore, introducing and studying Morita equivalence of symplectic groupoids should be the first step in understanding Morita equivalence of Poisson manifolds.