From symplectic groupoids to double structures

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.