Leaves of stacky Lie algebroids

Daniel Álvarez

doi:10.5802/crmath.37

Abstract

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliations. In particular, we show that for a (strict) Lie 2-group the coadjoint orbits of the units in the dual of its Lie 2-algebra are symplectic groupoids, meaning that the classical Kostant-Kirillov-Souriau symplectic forms on these special coadjoint orbits are multiplicative.

Highlights

Lie algebroids are a unifying concept in differential geometry: they allow us to describe foliations, Lie algebra actions, connections on principal bundles and Poisson brackets among many other examples
At least up to a connectedness issue, the leaves of the singular foliation associated to an LA-groupoid which pass through the unit manifold inherit themselves a Lie groupoid structure, see Theorem 6
We shall see in Proposition 12 that those leaves of the symplectic foliation associated to a Poisson groupoid structure which pass through the unit manifold provide examples of symplectic groupoids

Summary

Introduction

Lie algebroids are a unifying concept in differential geometry: they allow us to describe foliations, Lie algebra actions, connections on principal bundles and Poisson brackets among many other examples. At least up to a connectedness issue, the leaves of the singular foliation associated to an LA-groupoid which pass through the unit manifold inherit themselves a Lie groupoid structure, see Theorem 6. We shall see in Proposition 12 that those leaves of the symplectic foliation associated to a Poisson groupoid structure which pass through the unit manifold provide examples of symplectic groupoids. We exemplify this observation by means of (strict) Lie 2-groups [2], which are group objects in the category of Lie groupoids. The dressing action coincides with the coadjoint action on the dual of its Lie algebra so we get, in particular, that the coadjoint orbits of Lie 2-groups which contain a unit, endowed with their canonical Kostant–Kirillov–Souriau symplectic forms, are symplectic groupoids

Lie groupoids and Lie algebroids

LA-groupoids and double Lie groupoids

The main result

Symplectic leaves of Poisson groupoids

Orbits of Lie 2-algebra actions on Lie groupoids

Dressing orbits of Poisson 2-groups