Nambu-Dirac Structures on Lie Algebroids

TL;DR

This paper extends Nambu-Poisson structures to Lie algebroids using the Vinogradov bracket, introduces higher order Dirac structures, and unifies the description of these geometric structures within a common framework.

Contribution

It develops a natural extension of Nambu-Poisson structures to Lie algebroids and introduces higher order Dirac structures, unifying different geometric structures in this setting.

Findings

Nambu-Poisson structures on Lie algebroids are shown to be decomposable under certain conditions.

A new concept of higher order Dirac structures on Lie algebroids is introduced.

The framework unifies Nambu-Poisson and Dirac structures within Lie algebroids.

Abstract

The theory of Nambu-Poisson structures on manifolds is extended to the context of Lie algebroids, in a natural way based on the Vinogradov bracket associated with Lie algebroid cohomology. We show that, under certain assumptions, any Nambu-Poisson structure on a Lie algebroid is decomposable.Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu-Poisson structures on Lie algebroids and Dirac structures on manifolds in the same setting.