Nambu structures and associated bialgebroids

Abstract

We investigate 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.