Higher Extensions of Lie Algebroids

TL;DR

This paper explores higher extensions of Lie algebroids using representations up to homotopy, leading to new structures like higher Lie algebroids and applications in integrating Courant algebroids into Lie 2-groupoids.

Contribution

It introduces a framework for extending Lie algebroids with representations up to homotopy, resulting in higher Lie algebroids and their applications.

Findings

Construction of higher Lie algebroids from extensions

Examples include exact Courant algebroids and string Lie 2-algebras

Integration of exact Courant algebroids into Lie 2-groupoids

Abstract

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integration of an exact Courant algebroid.