Modular Classes of Lie Groupoid Representations up to Homotopy

TL;DR

This paper introduces a construction for the modular class of Lie groupoid representations up to homotopy, linking it to obstructions in volume form existence, thus advancing understanding of geometric structures on stacks.

Contribution

It provides a new method to define the modular class for representations up to homotopy of Lie groupoids, especially relating to the adjoint case and volume form obstructions.

Findings

Modular class construction for Lie groupoid representations up to homotopy.

Identification of the modular class as an obstruction to volume form existence.

Application to the adjoint representation and differentiable stacks.

Abstract

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack".