Nambu structures on Lie algebroids and their modular classes

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 and that of a Lie algebroid 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.