Separation of variables in the Jacobi identities

TL;DR

This paper introduces a broad family of solutions to the Jacobi identities, unifying various Poisson systems and enabling new global analytical methods.

Contribution

It characterizes a new, very general family of solutions to the Jacobi identities, linking diverse Poisson systems under a common framework.

Findings

Unifies many Poisson systems through a new solution family

Provides a conceptual framework for analyzing Poisson structures

Enables development of general global analysis methods

Abstract

A new family of $n$-dimensional solutions of the Jacobi identities is characterized. Such a family is very general, thus unifying in a common framework many different well-known Poisson systems seemingly unrelated. This unification is not only conceptual, but also allows the development of general global methods of analysis.