Algebraic Bethe ansatz for the elliptic quantum group $E_{\tau,\eta}(sl_n)$ and its applications

TL;DR

This paper develops an algebraic Bethe ansatz approach for the elliptic quantum group $E_{ au, ext{eta}}(sl_n)$, enabling exact solutions for related integrable models like the Belavin and Ruijsenaars-Schneider models.

Contribution

It introduces a nested Bethe ansatz method for diagonalizing transfer matrices of $E_{ au, ext{eta}}(sl_n)$ and applies it to solve specific elliptic integrable models.

Findings

Exact diagonalization of transfer matrices achieved.

Solutions provided for $Z_n$ Belavin model.

Solutions provided for elliptic $A_{n-1}$ Ruijsenaars-Schneider model.

Abstract

We study the tensor product of the {\it higher spin representations} (see the definition in Sect. 2.2) of the elliptic quantum group $E_{\tau,\eta}(sl_n)$. The transfer matrices associated with the $E_{\tau,\eta}(sl_n)$-module are exactly diagonalized by the nested Bethe ansatz method. Some special cases of the construction give the exact solution for the $Z_n$ Belavin model and for the elliptic $A_{n-1}$ Ruijsenaars-Schneider model.