Nambu-Dirac structures for Lie algebroids

The theory of Nambu-Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu-Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu-Poisson structures and Dirac structures on manifolds in the same setting.