Integrable lifts for transitive Lie algebroids

TL;DR

This paper demonstrates that the integrability obstruction for transitive Lie algebroids can be eliminated by adding extra dimensions, leading to new methods for constructing integrable lifts, including a universal de Rham-based approach.

Contribution

It introduces novel constructions of integrable lifts for transitive Lie algebroids, including a universal de Rham-based method and generalizations of Almeida-Molino lifts.

Findings

The Weinstein groupoid of a non-integrable transitive abelian Lie algebroid is a quotient of a finite-dimensional Lie groupoid.

A universal integrable algebroid can be constructed using the classical de Rham isomorphism.

Counterexamples to integrability can be generalized to create integrable lifts when the base manifold is simply connected.

Abstract

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid.