VB-groupoids and representation theory of Lie groupoids
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.
Cite
@article{arxiv.1007.3658,
title = {VB-groupoids and representation theory of Lie groupoids},
author = {Alfonso Gracia-Saz and Rajan Amit Mehta},
journal= {arXiv preprint arXiv:1007.3658},
year = {2017}
}
Comments
v5: Introduction is completely rewritten, many other improvements in the exposition. v6: Implements numerous corrections and changes suggested by referees, most notably a significant simplification of the calculations in Appendix A.2. Final version, to appear in J. Symp. Geom