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Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles
Camille Laurent-Gengoux1, Friedrich Wagemann2

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 is isomorphic to P. Neeb 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 paper, 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.
1:  LMA-Poitiers - Laboratoire de Mathématiques et Applications
2:  LMJL - Laboratoire de Mathématiques Jean Leray
crossed modules of Lie algebroids – crossed module of LIe groupoids – obstruction class – gerbe