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The First Pontryagin Class of a Quadratic Lie 2-Algebroid

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Abstract

In this paper, we first give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid T M to a strict Lie 3-algebroid constructed from derivations. Then, we introduce the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H5(M). Associated to a CLWX 2-algebroid, there is a quadratic Lie 2-algebroid naturally. Conversely, we show that the first Pontryagin class of a quadratic Lie 2-algebroid is the obstruction class of the existence of a CLWX-extension. Finally we construct a quadratic Lie 2-algebroid from a trivial principle 2-bundle with a \({\Gamma}\)-connection and show that its first Pontryagin class is trivial.

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Acknowledgements

We give our warmest thanks to Zhangju Liu, Konrad Waldorf, Xiaomeng Xu and Chenchang Zhu for very useful comments and discussions. We also give our special thanks to the referee for very helpful suggestions that improve the paper. This research is supported by NSFC (11471139) and NSF of Jilin Province (20170101050JC).

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  1. Department of Mathematics, Jilin University, Changchun, 130012, China

    Yunhe Sheng

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  1. Yunhe Sheng
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Correspondence to Yunhe Sheng.

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Communicated by C. Schweigert

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Sheng, Y. The First Pontryagin Class of a Quadratic Lie 2-Algebroid. Commun. Math. Phys. 362, 689–716 (2018). https://doi.org/10.1007/s00220-018-3220-y

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  • DOI: https://doi.org/10.1007/s00220-018-3220-y

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