Complete Lie algebroid actions and the integrability of Lie algebroids

TL;DR

This paper provides a new proof linking complete Lie algebroid actions on surjective submersions to their integrability, using double Lie groupoids and multiplicative foliations to characterize relevant structures.

Contribution

It introduces a novel proof connecting complete Lie algebroid actions with integrability, utilizing double Lie groupoids and multiplicative foliations.

Findings

New proof of the equivalence between complete actions and integrability

Characterization of vacant double Lie groupoids inducing Lie groupoid structures

Simplified approach using double Lie groupoids and foliations

Abstract

We give a new proof of the equivalence between the existence of a complete action of a Lie algebroid on a surjective submersion and its integrability. The main tools in our approach are double Lie groupoids and multiplicative foliations, our proof relies on a simple characterization of those vacant double Lie groupoids which induce a Lie groupoid structure on their orbit spaces.