On Nambu-Poisson manifolds.

*(English)*Zbl 0929.70015Nambu 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.

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.

Reviewer: M.de León (Madrid)

##### MSC:

70H99 | Hamiltonian and Lagrangian mechanics |

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

70H45 | Constrained dynamics, Dirac’s theory of constraints |

53Z05 | Applications of differential geometry to physics |

##### Keywords:

volume manifold; Hamiltonian vector field; Nambu-Poisson bracket; Casimir functions; first Lichnerowicz-Poisson cohomology group
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\textit{N. Nakanishi}, Rev. Math. Phys. 10, No. 4, 499--510 (1998; Zbl 0929.70015)

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##### References:

[1] | DOI: 10.1007/BF00312678 · Zbl 0861.70018 |

[2] | DOI: 10.1007/BF00400143 · Zbl 0849.70014 |

[3] | Nakanishi N., J. Math. Kyoto Univ. 31 pp 281– (1991) · Zbl 0724.53018 |

[4] | Nambu Y., Phys. Rev. 7 pp 2405– (1973) |

[5] | DOI: 10.1007/BF02103278 · Zbl 0808.70015 |

[6] | Weinstein A., J. Diff. Geom. 18 pp 523– (1983) |

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