VB-Groupoids and Representation Theory of Lie Groupoids

Authors

Alfonso Gracia-Saz, University of Toronto
Rajan Amit Mehta, Smith CollegeFollow

Document Type

Article

Publication Date

1-1-2017

Publication Title

Journal of Symplectic Geometry

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.

Volume

15

Issue

3

First Page

741

Last Page

783

DOI

10.4310/JSG.2017.v15.n3.a5

ISSN

15275256

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Rights

Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Gracia-Saz, Alfonso and Mehta, Rajan Amit, "VB-Groupoids and Representation Theory of Lie Groupoids" (2017). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/89