VB-groupoids and representation theory of Lie groupoids

TL;DR

This paper explores the relationship between VB-groupoids and 2-term representations up to homotopy of Lie groupoids, establishing a canonical cohomology theory and classifying regular 2-term representations.

Contribution

It introduces a correspondence between VB-groupoids and 2-term representations up to homotopy, and develops a canonical cohomology complex for VB-groupoids.

Findings

Tangent bundle corresponds to the adjoint representation.

Constructs a canonical cohomology complex for VB-groupoids.

Classifies regular 2-term representations up to homotopy.

Abstract

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint representation" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations up to homotopy. When applied to the tangent bundle of a Lie groupoid, this construction produces a canonical complex that computes the cohomology with values in the adjoint representation. Finally, we give a classification of regular 2-term representations up to homotopy. By considering the adjoint representation, we find a new cohomological invariant associated to regular Lie groupoids.