Lotka-Volterra systems as Poisson-Lie dynamics on solvable groups

A class of integrable 3D Lotka-Volterra (LV) equations is shown to be a particular instance of Poisson-Lie dynamics on a family of solvable 3D Lie groups. As a consequence, the classification of all possible Poisson-Lie structures on these groups is shown to provide a systematic approach to obtain multiparametric integrable deformations of this LV system. Moreover, by making use of the coproduct map induced by the group multiplication, a twisted set of 3N-dimensional integrable Lotka-Volterra equations can be constructed. Finally, the quantization of one of the Poisson-Lie LV structures is performed, and is shown to give rise to a quantum euclidean algebra.