Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

Abstract

Given a Lie-Poisson completely integrable bi-Hamiltonian system on $\mathbb{R}^n$, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group $G_η$ of dimension $n$, where $η\in \mathbb{R}$ is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on $G_η$ that underly the dynamics of the deformed system and by making use of the group law on $G_η$, one may obtain two completely integrable Hamiltonian systems on $G_η\times G_η$. By construction, both systems admit reduction, via the multiplication in $G_η$, to the deformed bi-Hamiltonian system in $G_η$. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.