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A note on \(n\)-ary Poisson brackets. (English) Zbl 0986.53035
Slovák, Jan (ed.) et al., The proceedings of the 19th Winter School “Geometry and physics”, Srní, Czech Republic, January 9-15, 1999. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 63, 165-172 (2000).
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 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 \(\{,\dots ,\}\) 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 trivially satisfied for any odd-order multivector \(P\). Therefore, in the odd case, the generalized Jacobi identity is unrelated with this condition.
In this paper, the authors obtain two conditions (an algebraic condition and a differential condition) which characterize when an \(n\)-vector (with \(n\) odd) defines an \(n\)-ary Poisson bracket. Moreover, they give an example of an \(n\)-ary structure of rank \(2n\), for any \(n\) (even or odd). These structures are defined by semi-decomposable \(n\)-vectors.
On the other hand, in [R. Weitzenböck, Invariantentheorie, P. Noordhoff, Groningen (1923; JFM 49.0064.01)] it is proved that an \(n\)-vector \(P\) is decomposable is and only if \(i(\alpha_1)i(\alpha_2)\dots i(\alpha_k)P\) is decomposable, for all covectors \(\alpha_1,\dots \alpha_k\), where \(k\in \{1,\dots ,n-2\}\) is fixed. The authors give an alternative proof of this result. As a consequence, they obtain that a \(3\)-vector \(P\) defines a ternary Poisson bracket on the manifold \(M\) if and only if, around every point \(x\in M\), \(P\) is decomposable. In particular, there is no differential condition for a Poisson \(3\)-vector since this condition is a consequence of decomposability.
For the entire collection see [Zbl 0940.00040].

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
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