Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

This paper is mainly a review of the multi-Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi-Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.