[Paper Review] A Note on n-ary Poisson Brackets

Peter W. Michor, Izu Vaisman|arXiv (Cornell University)|Jan 26, 1999

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper introduces a class of n-ary Poisson structures of constant rank and establishes that ternary Poisson brackets are precisely those defined by decomposable 3-vector fields. The key contribution is a lemma proving that an n-vector (n ≥ 3) is decomposable if and only if all its contractions with up to n−2 covectors are decomposable, providing a fundamental criterion for n-ary Poisson structures.

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.

Motivation & Objective

• To characterize a class of n-ary Poisson structures of constant rank.
• To determine the precise geometric condition under which ternary Poisson brackets arise.
• To establish a necessary and sufficient condition for the decomposability of n-vector fields (n ≥ 3).
• To provide a foundational lemma linking the decomposability of an n-vector to the decomposability of its contractions with covectors.
• To clarify the geometric structure underlying n-ary Poisson brackets beyond the binary case.

Proposed method

• Introduces n-ary Poisson brackets as multilinear, skew-symmetric brackets satisfying a generalized Jacobi identity.
• Analyzes the structure of n-vector fields and their decomposability in the context of Poisson geometry.
• Applies differential geometric techniques to study contractions of n-vectors with covectors.
• Proves a central lemma: an n-vector (n ≥ 3) is decomposable iff all its contractions with up to n−2 covectors are decomposable.
• Uses the lemma to characterize ternary Poisson brackets as those defined by decomposable 3-vector fields.
• Applies the result to show that constant rank n-ary Poisson structures are governed by decomposable n-vectors.

Experimental results

Research questions

• RQ1 What conditions ensure that an n-ary Poisson bracket arises from a multivector field?
• RQ2 When is a multivector field of degree n ≥ 3 decomposable?
• RQ3 What is the relationship between the decomposability of an n-vector and the decomposability of its contractions with covectors?
• RQ4 How can ternary Poisson brackets be geometrically characterized?
• RQ5 What is the role of the generalized Jacobi identity in n-ary Poisson structures of constant rank?

Key findings

• A class of n-ary Poisson structures of constant rank is explicitly identified.
• Ternary Poisson brackets are exactly those defined by decomposable 3-vector fields.
• The key lemma establishes that an n-vector (n ≥ 3) is decomposable if and only if all its contractions with up to n−2 covectors are decomposable.
• The result provides a geometric criterion for the existence of n-ary Poisson structures via multivector field decomposability.
• The characterization of ternary Poisson brackets via decomposable 3-fields offers a clear geometric interpretation.
• The work lays a foundation for studying higher-arity Poisson structures using multilinear algebra and differential geometry.