Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Camille Laurent-Gengoux Friedrich Wagemann

Institution:	(1) Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie, 86962 Futuroscope-Chasseneuil Cedex, France;(2) Laboratoire de Mathematiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 3, France

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.

Keywords:	Crossed modules of Lie algebroids Crossed modules of Lie groupoids Crossed modules of topological Lie algebras Obstruction class Bundle gerbe Deligne cohomology
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