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An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\dots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$ \sum_{\sigma\in S_{2n-1}}(\operatorname {sign}\sigma)\{\{f_{\sigma_1},\dots ,f_{\sigma_n}\},f_{\sigma_{n+1}},\dots ,f_{\sigma_{2n-1}}\}=0, $$ $S_{2n-1}$ being the symmetric group. The notion of generalized Poisson bracket was introduced by {\it J. A. de Azc\'arraga} 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 $\{,\dots ,\}$ 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! (English) |
