Picard groups in Poisson geometry

TL;DR

This paper investigates the structure of Picard groups in Poisson geometry, focusing on symplectic dual pairs and their Morita self-equivalences, with applications to rings, Lie groupoids, and symplectic groupoids.

Contribution

It introduces and analyzes the Picard group of a Poisson manifold via symplectic dual pairs and explores its variants in related algebraic and geometric contexts.

Findings

Characterization of the Picard group in Poisson geometry.

Examples illustrating the structure of Pic(P).

Extensions to rings, Lie groupoids, and symplectic groupoids.

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'' 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.