Integrable bi-Hamiltonian systems by Jacobi structure on real three-dimensional Lie groups

TL;DR

This paper develops a method to construct integrable bi-Hamiltonian systems on three-dimensional Lie groups by leveraging Jacobi structures and their Poissonization, expanding the toolkit for integrable systems in geometric contexts.

Contribution

It introduces a novel approach to derive integrable bi-Hamiltonian systems on Lie groups using Jacobi structures and their Poissonization, linking Lie algebra realizations to integrability.

Findings

Constructed integrable bi-Hamiltonian systems on Lie groups.

Demonstrated the use of Jacobi structures in integrability.

Provided explicit realizations of Lie algebra structures.

Abstract

By Poissonization of Jacobi structures on real three-dimensional Lie groups $\mathbf{G}$ and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on $\mathbf{G}\otimes \mathbb{R}$.