New solutions of the Jacobi equations for three-dimensional Poisson structures

TL;DR

This paper systematically classifies three-dimensional Poisson structures by solving Jacobi equations, unifying many known structures into broader families, and developing algorithms for their analysis that are globally valid and more efficient.

Contribution

It introduces three new families of solutions to the Jacobi equations, unifying various Poisson structures and enabling global analysis algorithms.

Findings

Unified framework for Poisson structures

Algorithms for symplectic and Casimir invariants

Global validity of Darboux forms

Abstract

A systematic investigation of the skew-symmetric solutions of the three-dimensional Jacobi equations is presented. As a result, three disjoint and complementary new families of solutions are characterized. Such families are very general, thus unifying many different and well-known Poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution. This unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure, the Casimir invariants and the Darboux canonical form, which are known only for a limited sample of Poisson structures. These common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework, instead of using a case-by-case approach. In addition, the methods developed are valid globally in phase space, thus ameliorating the usual scope of Darboux' reduction which is only of local nature. Finally, the families of solutions found present some new nonlinear superposition principles which are characterized.