Nnambu–Poisson structures and their foliations

Nambu–Poisson bracket is a natural generalization of Poisson bracket. A very distinguished property is its decomposability. This is investigated from the second order term of the fundamental identity (see [2] or [5]). In this paper, we shall study the first order term of the fundamental identity and get a relation with the Schouten–Nijenhuis bracket. And also we shall show that for a given Poisson structure, the top power of it gives a Nambu–Poisson structure. We shall characterize the Godbillon–Vey class of the foliation defined from a regular Nambu–Poisson tensor.