Picard groups of Poisson manifolds

TL;DR

This paper develops methods to compute the Picard group of Poisson manifolds, revealing its structure and relationship with gauge transformations, and proves a conjecture relating Picard groups of Lie algebra duals to outer automorphisms.

Contribution

It introduces systematic techniques for computing Picard groups of Poisson manifolds and establishes their connection with gauge transformations and automorphism groups.

Findings

Computed Picard groups for specific Poisson manifolds.

Proved the conjecture relating Picard groups of Lie algebra duals to outer automorphisms.

Established the equivalence of connected components of the identity in Picard groups and gauge transformations.

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 $\mathfrak{g}$.