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arXiv · 1807.11789

The first Pontryagin class of a quadratic Lie 2-algebroid

Yunhe Sheng
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Abstract

In this paper, first we 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 TM 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 H^5(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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BibTeXRIS

Yunhe Sheng. 2018-07-31. The first Pontryagin class of a quadratic Lie 2-algebroid. https://doi.org/10.1007/s00220-018-3220-y

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