Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

TL;DR

This paper explores the structure of VB-algebroids as Lie algebroid objects in vector bundle categories, establishing a correspondence with superconnections, classifying regular cases, and introducing new characteristic classes.

Contribution

It provides a complete classification of regular VB-algebroids and introduces a novel characteristic class not seen in traditional Lie algebroid representation theory.

Findings

Established a one-to-one correspondence between VB-algebroids and flat superconnections.

Constructed characteristic classes that generalize those by Crainic and Fernandes.

Classified all regular VB-algebroids and identified a new characteristic class.

Abstract

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.