An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\cdots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$\sum _{\sigma \in S_{2n-1}}(sign\sigma )\{\{f_{{\sigma }_1},\cdots ,f_{{\sigma }_n}\},f_{{\sigma }_{n+1}},\cdots ,f_{{\sigma }_{2n-1}}\}=0,$$ $S_{2n-1}$ being the symmetric group. The notion of generalized Poisson bracket was introduced by J. A. de Azcárraga et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]. They established that an $n$-ary Poisson bracket $\{,\cdots ,\}$ defines an $n$-vector $P$ on the manifold $M$ such that, for $n$ even, the generalized Jacobi identity is translated by the equation $[P,P]=0,$ where $[,]$ is the Schouten-Nijenhuis bracket. When $n$ is odd, the condition $[P,P]=0$ is!

EUDML-ID : urn:eudml:doc:220511
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@article{701659,
     title = {A note on n-ary Poisson brackets},
     booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2000},
     pages = {165-172},
     mrnumber = {MR1758092},
     zbl = {0986.53035},
     url = {http://dml.mathdoc.fr/item/701659}
}

Michor, Peter W.; Vaisman, Izu. A note on n-ary Poisson brackets, dans Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books,  (2000), pp. 165-172. http://gdmltest.u-ga.fr/item/701659/