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