Picard groups in Poisson geometry

Abstract

We study isomorphism classes of symplectic dual pairs P <- S -> P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural ``tensor product&#39;&#39; operation. We discuss this group in several examples and study variants of this construction for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.