Regular Cartan groupoids and longitudinal representations

TL;DR

This paper explores the relationship between Cartan connections on proper regular Lie groupoids and their longitudinal representations, simplifying the problem and providing examples of obstructions to multiplicative connections.

Contribution

It establishes a link between Cartan connections and longitudinal representations, reducing the problem to a more manageable form and analyzing obstructions in rank two cases.

Findings

Relationship between Cartan connections and longitudinal representations clarified

Reduction of the problem to a simpler one demonstrated

Examples of obstructions in rank two cases provided

Abstract

With the intent of laying the groundwork for a program that aims at explicitly describing the space of Cartan (i.e. multiplicative) connections on a general proper Lie groupoid, we begin to investigate the space of such connections in the regular case. We point out that there is a close relationship between Cartan connections on a proper regular groupoid and representations of the groupoid on its own longitudinal bundle (i.e. on the vector distribution tangent to its orbits). This observation enables us to reduce the original problem to a simpler one. We carry out a prospective study of the latter problem, and apply the resulting analysis to produce a number of examples in rank two which serve to illustrate the diversity of the possible obstructions to the existence of multiplicative connections.