Non-Decomposable Nambu Brackets

TL;DR

This paper introduces a weaker condition than the Fundamental Identity that permits non-decomposable Nambu brackets while still enabling a Darboux-like theorem, expanding the understanding of Nambu structures.

Contribution

It proposes a new condition weaker than the FI, allowing non-decomposable Nambu brackets and establishing a Darboux-like theorem for them.

Findings

Weaker condition than FI enables non-decomposable brackets.

Darboux-like theorem extended to these generalized brackets.

Generalization of Weinstein's splitting principle for Nambu manifolds.

Abstract

It is well-known that the Fundamental Identity (FI) implies that Nambu brackets are decomposable, i.e., given by a determinantal formula. We find a weaker alternative to the FI that allows for non-decomposable Nambu brackets, but still yields a Darboux-like Theorem via a Nambu-type generalization of Weinstein's splitting principle for Poisson manifolds.