Van Est isomorphism for homogeneous cochains
Differential Geometry
2017-09-27 v1 Representation Theory
Abstract
VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case k=1 yields a Van Est map for representations up to homotopy on 2-term graded vector bundles. Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van Est theorem for differential forms on Lie groupoids with values in a representation.
Cite
@article{arxiv.1602.06887,
title = {Van Est isomorphism for homogeneous cochains},
author = {Alejandro Cabrera and Thiago Drummond},
journal= {arXiv preprint arXiv:1602.06887},
year = {2017}
}