Lie Algebroids, Holonomy and Characteristic Classes

TL;DR

This paper extends the concept of connections to Lie algebroids, enabling the study of holonomy, stability, and new characteristic classes that generalize existing invariants.

Contribution

It introduces a generalized notion of connection on Lie algebroids, defines holonomy for their orbit foliations, and constructs new characteristic classes extending the modular class.

Findings

Defined holonomy of Lie algebroid orbit foliations

Proved a stability theorem for these structures

Introduced secondary characteristic classes that generalize the modular class

Abstract

We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit foliation of a Lie algebroid and prove a stability theorem. We also introduce secondary or exotic characteristic classes that generalize the modular class of a Lie algebroid.