Nambu brackets were introduced by Y. Nambu in 1973, but it seems that the subject comes back to M. L. Albeggiani (see [E. T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies. With an introduction to the problem of three bodies. With a foreword by Sir William McCrea. Repring of the fourth edition. Cambridge University Press. xvii (1988; Zbl 0665.70002), p. 337]). In any case, the motivation for Nambu was to generalize Hamiltonian mechanics. Since the publication of Nambu’s article, a lot of papers were published trying to relate these multibrackets with Dirac theory of constraints, to quantize them, and to discuss other problems. Later, in 1994, the subject experienced a new impulse after a paper by L. Takhtajan [Commun. Math. Phys. 160, No. 2, 295-315 (1994; Zbl 0808.70015)], who introduced the notion of Nambu-Poisson manifold by adding an integrability condition called fundamental identity. The geometric structure of Nambu-Poisson manifolds was clarified by several authors: P. Gautheron, D. Alekseevsky, P. Guha, J. Grabowski, G. Marmo, I. Vaisman, R. Ibáñez, D. Martín de Diego, J. C. Marrero and the reviewer.
In the paper under review the author gives an elegant proof of the local structure of a Nambu-Poisson manifold, that is, at a regular point, it is a local product of a volume manifold by a trivial structure. The author also studies the notions of Hamiltonian vector field and Casimir “function”, and introduces a bracket in \(\Lambda^{n-1}{\mathcal F}\), where \(n\) is the order of the Nambu-Poisson bracket, and \({\mathcal F}\) is the ring of \(C^\infty\)-function on \(M\). This bracket is not skew-symmetric, but it is so when projected to the quotient by the space of Casimir functions. Also, it is proved that the space \({\mathcal H}\) of Hamiltonian vector fields is an ideal of the space \({\mathcal L}\) of infinitesimal automorphisms of the Nambu-Poisson structure. The author studies the quotient \({\mathcal L}/{\mathcal H}\) which in Poisson manifolds is just the first Lichnerowicz-Poisson cohomology group. He claims that there is no a convenient notion of such a cohomology for Nambu-Poisson manifolds. Indeed, R. Ibáñez, J. C. Marrero and the reviewer have recently introduced a generalized Poisson cohomology [J. Phys. A, Math. Gen. 31, No. 4, 1253-1266 (1998; Zbl 0907.58017)], but it seems not to be more adequate in order to classify Nambu-Poisson structures. A different approach is given by R. Ibáñez, J. C. Marrero, E. Padrón and the reviewer [Leibniz algebroid associated with a Nambu-Poisson structure, Preprint math-ph/9906027), where a Leibniz algebroid is naturally associated to each Nambu-Poisson manifold. Its cohomology permits to introduce the notion of modular class as in the case of Poisson manifolds.