Ja n 19 99 A Note on n - ary Poisson Brackets

A class of n-ary Poisson structures of constant rank is indicated. Then, one proves that the ternary Poisson brackets are exactly those which are defined by a decomposable 3-vector field. The key point is the proof of a lemma which tells that an n-vector (n ≥ 3) is decomposable iff all its contractions with up to n − 2 covectors are decomposable. In the last years, several authors have studied generalizations of Lie algebras to various types of n-ary algebras, e.g., [5, 11, 8, 10, 14]. In the same time, and intended to physical applications, the new types of algebraic structures were considered in the case of the algebra C(M) of functions on a C manifold M , under the assumption that the operation is a derivation of each entry separately. In this way one got the Nambu-Poisson brackets, e.g., [11, 6, 1, 4], and the generalized Poisson brackets [2, 3], etc. In this note, we write down the characteristic conditions of the n-ary generalized Poisson structures in a new form, and give an example of an n-ary structure of constant rank 2n, for any n even or odd. Then, we prove that the ternary Poisson brackets are exactly the brackets defined by the decomposable 3vector fields. The key point in the proof of this result is a lemma (that seems to appear also in [15]), which tells that an n-vector P is decomposable iff ∗1991 Mathematics Subject Classification 58 F 05.