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

Given a Lie-Poisson completely integrable bi-Hamiltonian system on Rn, 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 Gη of dimension n, where η∈R is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on Gη that underly the dynamics of the deformed system and by making use of the group law on Gη, one may obtain two completely integrable Hamiltonian systems on Gη×Gη. By construction, both systems admit reduction, via the multiplication in Gη, to the deformed bi-Hamiltonian system in Gη. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.