Nambu Structures And Associated Bialgebroids

TL;DR

This paper explores higher order generalizations of Lie algebroids and bialgebroids, establishing their connections to Nambu-Poisson structures and introducing weak Lie-Filippov bialgebroids, with applications to Nambu Lie groups.

Contribution

It introduces the concept of weak Lie-Filippov bialgebroids as a higher order generalization and links them to Nambu-Poisson structures and Nambu Lie groups.

Findings

$n$-Lie algebroid structures correspond to $n$-ary Gerstenhaber algebras.

Nambu-Poisson manifolds induce weak Lie-Filippov bialgebroids.

Tangent spaces of Nambu Lie groups form examples of weak Lie-Filippov bialgebroids.

Abstract

This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.