Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Abstract

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension \(1\to Z\to \hat{K}\to K\to 1\) of K. It is a classical question whether there exists a \(\hat{K}\) -principal bundle \(\hat{P}\) on M such that \(\hat{P}/Z\cong P\) . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class \([\omega_{\rm top\,\,alg}]\) is an obstruction to the existence of \(\hat{P}\) . In the present article, we show that \([\omega_{\rm top\,\,alg}]\) is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.