Ruijsenaars' commuting difference operators as commuting transfer matrices

Abstract

For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation $RLL=LLR$, the trace of the L-operator gives a commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix as well as other approaches for the elliptic commuting system is given. We also study the invariant subspace for the system which is spanned by symmetric theta functions.