Non-commutative and Commutative Integrability of Generic Toda Flows in Simple Lie Algebras

1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s variables [F] the finite non-periodic Toda lattice describes an isospectral evolution on the set of tri-diagonal matrices in sl(n). It was explicitly solved and shown to be completely integrable by Moser [Mo] . In his paper [K1], Kostant comprehensively studied the generalization of Toda lattice that evolves on the set of “tri-diagonal” elements of a semisimple Lie algebra g which also turned out to be completely integrable with Poisson commuting integrals being provided by the Chevalley invariants of the algebra. Moreover, in this paper, as well as in the works by Ol’shanetsky and Perelomov [OP], Reyman and Semenov-Tian-Shansky [RSTS1], Symes [Sy], the method of the explicit integration of the Toda equations was extended to the case when evolution takes place on the dual space of the Borel subalgebra of g . This space is foliated into symplectic leaves of different dimensions and the natural question is what can be said about the Liouville complete integrability of the Toda flows on each of these leaves. In the particular case of generic symplectic leaves in sl(n) the complete integrability was proved by Deift, Li, Nanda and Tomei [DLNT]. This paper was motivated by the work [DLNT] and its Lie algebraic interpretation proposed in [S1], [S2], [EFS]. Our main result is the following