The transverse density bundle and modular classes of Lie groupoids

Abstract

In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle DAtr instead of QA[jls-end-space/], which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, QA is not really an object associated with the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).We provide a simple construction of the representation of a Lie groupoid on its transverse volume, orientation, and density bundles in terms of (good) functors on vector spaces. We also extend the modular class by a Stiefel-Whitney class that controls the transverse orientability of a Lie groupoid.