The first Pontryagin class of a quadratic Lie 2-algebroid

TL;DR

This paper studies quadratic Lie 2-algebroids, introduces their first Pontryagin class as a cohomology class, and explores its role as an obstruction to CLWX-extension, with applications to trivial principle 2-bundles.

Contribution

It defines the first Pontryagin class for quadratic Lie 2-algebroids and links it to the existence of CLWX-extensions, providing new insights into their structure and obstructions.

Findings

The first Pontryagin class is a cohomology class in H^5(M).

The class acts as an obstruction to CLWX-extension.

Constructed a quadratic Lie 2-algebroid from a trivial principle 2-bundle.

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.