Regular XXZ Bethe states at roots of unity – as highest weight vectors of the sl 2 loop

We show that every regular Bethe ansatz eigenstate of the XXZ spin chain at roots of unity is a highest weight vector of the sl2 loop algebra and discuss whether it generates an irreducible representation or not. We show it in some sectors with respect to eigenvalues of the total spin operator SZ . The parameter q is given by a root of unity, q2N 0 = 1, for an integer N . Here, q is related to the XXZ coupling ∆ by ∆ = (q + q−1)/2. 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 state regular if it has regular Bethe roots. In the proof we assume that any set of regular Bethe roots at q0 gives an isolated solution of the Bethe ansatz equations. We evaluate explicitly the highest weight of a regular Bethe state in the sectors, and introduce parameters expressing it. If the parameters are distinct, the regular Bethe state generates an irreducible representation and we obtain the Drinfeld polynomial. If they are not distinct, however, it does not necessarily generate an irreduciblle one. We show such a regular Bethe state in the inhomogeneous case that generates a four-dimensional reducible representation.