Deformations of Vector Bundles over Lie Groupoids

TL;DR

This paper studies the deformations of VB-groupoids, a class of geometric objects related to Lie groupoids, by developing a cochain complex framework and exploring invariance properties and applications.

Contribution

It introduces a cochain complex for controlling VB-groupoid deformations and analyzes fundamental features like Morita invariance and a van Est theorem.

Findings

Cochain complex effectively controls VB-groupoid deformations

Morita invariance of the deformation complex established

Applications to classical objects and geometric interpretations

Abstract

VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.