From symplectic groupoids to double structures

TL;DR

This paper introduces symplectic groupoids and their associated double structures, bridging classical and categorical approaches, with a focus on new perspectives and simplified explanations for readers familiar with basic symplectic geometry and Lie theory.

Contribution

It presents a novel approach to symplectic groupoids and double structures, combining classical and categorical methods with some new features for clearer understanding.

Findings

Main results are known but presented with new approach

Highlights categorical compatibility conditions in Poisson groupoids

Provides a quick introduction for readers with basic symplectic and Lie theory knowledge

Abstract

These notes are an introduction to symplectic groupoids and the double structures associated with them. The treatment is intended to lie about midway between the original account of Coste, Dazord and Weinstein, which relied on effective use of the symplectic structures, and the account in my 2005 book, which showed, on the level of Poisson groupoids, that the basic results of the theory follow from `categorical' compatibility conditions between the associated Lie algebroid and Lie groupoid structures. The reader needs to know only the most basic ideas of symplectic geometry, Lie groups, and vector bundles. These notes aim to introduce the reader to certain important, and relatively new, ideas quickly; accordingly they omit much standard material. All the main results here are known, but the approach has a few new features.