Non-compact symmetric spaces and the Toda molecule equations

Abstract

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, GN, by its maximal compact subgroup. This is explained in more detail and it is shown that the “fundamental Poisson bracket relation” involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forGN endows the symmetric space with a hidden group theoretic structure.