Picard Groups of Poisson Manifolds

For a Poisson manifold M we develop systematic methods to compute its Picard group Pic(M), i.e., its group of self Morita equivalences. We establish a precise relationship between Pic(M) and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of M and to study its basic properties. Our methods lead to the proof of a conjecture from [3] stating that Pic(g∗) for any compact simple Lie algebra agrees with the group of outer automorphisms of g.