zbMATH — the first resource for mathematics

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
Keywords:
$$n$$-ary Poisson brackets
Full Text: