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.