# 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!

## How to cite

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Michor, Peter W., and Vaisman, Izu. "A note on n-ary Poisson brackets." Proceedings of the 19th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 2000. 165-172. <http://eudml.org/doc/220511>.

@inProceedings{Michor2000,
abstract = {An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\lbrace ,\dots ,\rbrace$ 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 )\lbrace \lbrace f\_\{\sigma \_1\},\dots ,f\_\{\sigma \_n\}\rbrace ,f\_\{\sigma \_\{n+1\}\},\dots ,f\_\{\sigma \_\{2n-1\}\}\rbrace =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 $\lbrace ,\dots ,\rbrace$ 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!},
author = {Michor, Peter W., Vaisman, Izu},
booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {165-172},
publisher = {Circolo Matematico di Palermo},
title = {A note on n-ary Poisson brackets},
url = {http://eudml.org/doc/220511},
year = {2000},
}

TY - CLSWK
AU - Michor, Peter W.
AU - Vaisman, Izu
TI - A note on n-ary Poisson brackets
T2 - Proceedings of the 19th Winter School "Geometry and Physics"
PY - 2000
CY - Palermo
PB - Circolo Matematico di Palermo
SP - 165
EP - 172
AB - An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\lbrace ,\dots ,\rbrace$ 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 )\lbrace \lbrace f_{\sigma _1},\dots ,f_{\sigma _n}\rbrace ,f_{\sigma _{n+1}},\dots ,f_{\sigma _{2n-1}}\rbrace =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 $\lbrace ,\dots ,\rbrace$ 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!
KW - Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/220511
ER -

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