Modular class of Lie $\infty$-algebroids and adjoint representation

TL;DR

This paper investigates the modular class of negatively graded Lie $ $-algebroids, demonstrating its equivalence across various descriptions, its homotopy invariance, and explicitly detailing the adjoint and coadjoint actions up to homotopy.

Contribution

It introduces a unified view of the modular class for Lie $ $-algebroids, clarifies its properties, and provides explicit dualities and actions up to homotopy.

Findings

Modular class equivalence across descriptions

Homotopy invariance of the modular class

Explicit formulations of adjoint and coadjoint actions

Abstract

We study the modular class of $Q$-manifolds, and in particular of negatively graded Lie $\infty$-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie $\infty$-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie $\infty$-algebroids and their $Q$-manifold equivalent, which we hope to be of use for future reference.