Regular XXZ Bethe states at roots of unity as highest weight vectors of the sl2 loop algebra

Abstract

We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator SZ, and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots. We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state.