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

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

Authors:	Camille Laurent-Gengoux Friedrich Wagemann

Affiliation:	(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 of K. It is a classical question whether there exists a -principal bundle on M such that . 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 . 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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