Bi-Hamiltonian Formulation of Generalized Toda Chains

In this paper we establish for the first time the bi-Hamiltonian nature of the Toda lattices corresponding to classical simple Lie groups. These are systems that generalize the usual finite, non-periodic Toda lattice (which corresponds to a root system of type An). This generalization is due to Bogoyavlensky [2]. These systems were studied extensively in [10] where the solution of the system was connected intimately with the representation theory of simple Lie groups. There are also studies by Adler, van Moerbeke [1] and Olshanetsky, Perelomov [15]. We call such systems the Bogoyavlensky–Toda lattices. We make the following more general definition which involves systems with exponential interaction: Consider a Hamiltonian of the form