Modular classes of Lie algebroid morphisms

TL;DR

This paper explores how the modular class of a Lie algebroid transforms under various morphisms, introducing the concept of the relative modular class and analyzing its behavior in different contexts.

Contribution

It introduces the relative modular class for Lie algebroid morphisms and studies its properties, including its behavior under pull-back, base-preserving morphisms, and generalized morphisms like Morita equivalences.

Findings

The relative modular class is a coboundary in the category of Lie algebroids and generalized morphisms.

Modular classes of pull-back and extension morphisms are characterized.

Generalized morphisms act on the 1-cohomology, affecting modular classes.

Abstract

We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.