A Note on n-ary Poisson Brackets

TL;DR

This paper characterizes n-ary Poisson brackets of constant rank, showing that ternary brackets are precisely those defined by decomposable 3-vectors, with a key lemma relating decomposability to contractions.

Contribution

It provides a new characterization of n-ary Poisson brackets and proves a lemma linking decomposability of n-vectors to their contractions.

Findings

Ternary Poisson brackets are exactly those from decomposable 3-vectors.

A key lemma relates decomposability of n-vectors to contractions with covectors.

The paper introduces a class of n-ary Poisson structures of constant rank.

Abstract

A class of n-ary Poisson structures of constant rank is indicated. Then, one proves that the ternary Poisson brackets are exactly those which are defined by a decomposable 3-vector field. The key point is the proof of a lemma which tells that an n-vector $(n\geq3)$ is decomposable iff all its contractions with up to n-2 covectors are decomposable.