Multiplicative Nambu structures on Lie groupoids

TL;DR

This paper introduces Nambu-Lie groupoids, generalizing Poisson-Lie groupoids, and explores their properties, including coisotropic submanifolds and subgroupoids, extending classical results to multivector fields.

Contribution

It defines Nambu-Lie groupoids, generalizes Weinstein's results to multivector fields, and links subgroupoids to coisotropic subalgebroids, broadening the theory of Lie groupoids.

Findings

Introduction of Nambu-Lie groupoids and their properties

Generalization of Weinstein's results to Nambu-Poisson tensors

Establishment of correspondence between coisotropic subgroupoids and coisotropic subalgebroids

Abstract

We study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field. Using this notion, we generalize the results of Weinstein \cite{wein} from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. We also introduce the notion of Nambu-Lie groupoid generalizing the concepts of both Poisson-Lie groupoid and Nambu-Lie group. We show that the infinitesimal version of Nambu-Lie groupoid is the notion of weak Lie-Filippov bialgebroid as introduced in \cite{bas-bas-das-muk}. Next we introduce coisotropic subgroupoids of a Nambu-Lie groupoid and these subgroupoids corresponds to, so called coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid.