Leaves of stacky Lie algebroids

TL;DR

This paper investigates the structure of leaves in LA-groupoids, revealing they are closely related to Lie groupoids, and explores their applications in symplectic groupoids derived from Poisson groupoids and Lie 2-groups.

Contribution

It demonstrates that leaves passing through the unit manifold in LA-groupoids are essentially Lie groupoids and connects this to symplectic structures in Poisson and Lie 2-group contexts.

Findings

Leaves passing through the unit manifold are Lie groupoids.

Coadjoint orbits in Lie 2-algebras are symplectic groupoids.

Classical symplectic forms are multiplicative on these orbits.

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.