A note on n-ary Poisson brackets

Michor, Peter W. ; Vaisman, Izu

Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books, (2000), p. 165-172 / Harvested from

Résumé

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., $$\sum _{\sigma \in S_{2n-1}}(sign\sigma )\{\{f_{{\sigma }_1},\cdots ,f_{{\sigma }_n}\},f_{{\sigma }_{n+1}},\cdots ,f_{{\sigma }_{2n-1}}\}=0,$$ $S_{2n-1}$ 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 $\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!

Détail

MR MR1758092 | Zbl 0986.53035

EUDML-ID : urn:eudml:doc:220511
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