Vector bundles over Lie groupoids and algebroids

TL;DR

This paper explores the theory of vector bundles over Lie groupoids and algebroids, clarifying their relationships through differentiation and integration, and extending techniques to more complex structures like double Lie algebroids.

Contribution

It provides a new reformulation of VB-groupoids and VB-algebroids, elucidates their Lie theory relations, and extends methods to double Lie algebroids and LA-groupoids.

Findings

Reformulated definitions of VB-groupoids and VB-algebroids

Clarified their relation via Lie theory

Extended techniques to double Lie algebroids and LA-groupoids

Abstract

We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their relation via differentiation and integration. We also show how to extend our techniques to describe the more general Lie theory underlying double Lie algebroids and LA-groupoids.