A note on n-ary Poisson brackets

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 -