# A note on n-ary Poisson brackets

• Proceedings of the 19th Winter School "Geometry and Physics", Publisher: Circolo Matematico di Palermo(Palermo), page 165-172

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## Abstract

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An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\left\{,\cdots ,\right\}$ 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}}\left(sign\sigma \right)\left\{\left\{{f}_{{\sigma }_{1}},\cdots ,{f}_{{\sigma }_{n}}\right\},{f}_{{\sigma }_{n+1}},\cdots ,{f}_{{\sigma }_{2n-1}}\right\}=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 $\left\{,\cdots ,\right\}$ defines an $n$-vector $P$ on the manifold $M$ such that, for $n$ even, the generalized Jacobi identity is translated by the equation $\left[P,P\right]=0,$ where $\left[\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\right]$ is the Schouten-Nijenhuis bracket. When $n$ is odd, the condition $\left[P,P\right]=0$ is!

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