The first Pontryagin class of a quadratic Lie 2-algebroid

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.

28 pages

References in corpus (5)

• On the category of Lie n-algebroids
• Principal infinity-bundles -- General theory
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• Courant Algebroids. A Short History
• QP-Structures of Degree 3 and 4D Topological Field Theory

Cited by in corpus (1)

• Categorification of VB-Lie algebroids and VB-Courant algebroids