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Nambu structures and associated bialgebroids. (English) Zbl 07006406
Summary: 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.
##### MSC:
 17B62 Lie bialgebras; Lie coalgebras 17B63 Poisson algebras 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D17 Poisson manifolds; Poisson groupoids and algebroids
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