Van Est isomorphism for homogeneous cochains

TL;DR

This paper extends the Van Est isomorphism to homogeneous cochains on VB-groupoids, providing new tools for studying representations of Lie groupoids and algebroids with applications to differential forms and homotopy representations.

Contribution

It introduces a Van Est theorem for k-homogeneous cochains on VB-groupoids, refining the cohomology theory and enabling new applications in representation theory.

Findings

Established Van Est isomorphism for homogeneous cochains

Derived Van Est map for representations up to homotopy

Developed Van Est theorem for differential forms with values in representations

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.