On the bi-Hamiltonian structure of Bogoyavlensky-Toda lattices

In this paper we prove that the Bogoyavlensky-Toda lattices corresponding to simple, classical Lie groups are bi-Hamiltonian. A review of previous results on the multi-Hamiltonian structure of these systems is also included. The areas investigated include master symmetries, recursion operators, invariants and higher Poisson brackets. The case of exceptional, simple Lie groups is still an open problem.