On models for the adjoint representation of a Lie algebroid

From CRCG-Wiki

The usual notion of a Lie algebroid representation is known to be inadequate, in the sense that it does not include a natural adjoint representation. In order to circumvent this problem, various authors have proposed generalizations of the definition of representation, including representations up to homotopy (Evens-Lu-Weinstein), basic connections (Fernandes), Lie algebroid modules (Vaintrob), and, most recently, VB-algebroids (joint work with A. Gracia-Saz). I will describe the various notions and the similarities and differences between them. Then I will describe the classification of VB-algebroids and a strange characteristic class that emerges from the analysis.