Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

TL;DR

This paper introduces a new method for deriving compatible Poisson structures on Lie groups using adjoint representations, leading to the discovery of new bi-Hamiltonian systems on specific symplectic Lie groups.

Contribution

It presents an alternative approach to find compatible Poisson structures and constructs new integrable bi-Hamiltonian systems on certain Lie groups.

Findings

Derived compatible Poisson structures on specific Lie groups.

Constructed new bi-Hamiltonian systems using Magri-Morosi's theorem.

Extended the understanding of integrable systems on symplectic Lie groups.

Abstract

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.