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

TL;DR

This paper introduces a new four-dimensional family of solutions to the Jacobi equations for Poisson structures, unifying various known structures and enabling explicit analysis of their properties globally in phase space.

Contribution

It presents a novel family of solutions to the Jacobi equations for Poisson structures, extending the scope of analysis and applicability beyond known cases.

Findings

New four-dimensional family of Poisson solutions characterized

Known Poisson structures are special cases of the new family

Explicit methods for Casimir invariants and symplectic structures provided

Abstract

A new four-dimensional family of skew-symmetric solutions of the Jacobi equations for Poisson structures is characterized. As a consequence, previously known types of Poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of the new family of solutions. Additionally, it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions, such as their Casimir invariants, symplectic structure and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a limited sample of known finite-dimensional Poisson structures. Moreover, the results developed are valid globally in phase space, thus ameliorating the usual scope of Darboux theorem which is of local nature.