Nambu structures on Lie algebroids and their modular classes

TL;DR

This paper defines the modular class for Lie algebroids with Nambu structures, extending known properties from Nambu-Poisson manifolds and linking them to existing modular classes of Lie algebroids.

Contribution

It introduces the modular class concept for Lie algebroids with Nambu structures and explores its properties and relations to existing modular classes.

Findings

Modular class of Nambu-Poisson manifolds is recovered as a special case.

Properties of modular classes extend from Nambu-Poisson manifolds to Lie algebroids.

Modular classes relate to Evens-Lu-Weinstein classes for certain Nambu-Poisson manifolds.

Abstract

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $\Lambda$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $\Lambda.$ We show that many known properties of the modular class of a Nambu-Poisson manifold that of a Lie algebropid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens-Lu-Weinstein modular classes of Lie algebroids.