Algebraic Bethe ansatz for the elliptic quantum group $E_{\tau,\eta}(sl_2)$

Abstract

To each representation of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a $q$-deformation of Hermite's solution of the Lam\'e equation.