Picard groups of Poisson manifolds

Abstract

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, in particular, to the proof of a conjecture from [BW04] stating that for any compact simple Lie algebra $\mathfrak{g}$ the group $Pic(\mathfrak{g}^*)$ concides with the group of outer automorphisms of $\mathfrak{g}$.