The finite Toda lattices
A potentially fruitful field of research is opened by this paper in which the author uses his previous results that multiparticle systems in which the structure of an interaction potential is defined by the systems of all positive, and not just simple roots of classical Lie algebras are also completely integrable. Flaschka and Moser have proved complete integrability of finite dimensional Hamiltonian systems of that kind (nonperiodic and periodic Toda lattices). Kostant, Olshanetsky and Perelomov have established the connection between nonperiodic lattices and Lie algebras. The author also shows that the above-listed Toda lattices are particular cases of these systems corresponding to certain limit situations. And taking advantage of that correspondence, the author succeeded in obtaining new integrable lattices differing from the known ones by the interaction of particles near the ends. Explicit expressions for the Lax matrices of the new lattices and non-Abelian generalization of some of them are presented at the end of the paper. They are too complicated to reproduce them here.
scientific article; zbMATH DE number 4109268