Spin generalization of the Ruijsenaars-Schneider model, the non-Abelian 2D Toda chain, and representations of the Sklyanin algebra

Abstract

Action-angle type variables for spin generalizations of the elliptic\nRuijsenaars-Schneider system are constructed. The equations of motion of these\nsystems are solved in terms of Riemann theta-functions. It is proved that these\nsystems are isomorphic to special elliptic solutions of the non-abelian 2D Toda\nchain. A connection between the finite gap solutions of solitonic equations and\nrepresentations of the Sklyanin algebra is revealed and discrete analogs of the\nLame operators are introduced. A simple way to construct representations of the\nSklyanin algebra by difference operators is suggested.\n