Bethe states as highest weight vectors of the sl 2 loop algebra at roots of unity

We show that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unityare highest weight vectors and generate irreducible representations of the sl2 loop algebra.Here the parameter q, which is related to the XXZ anisotropy ∆ through ∆ = (q+q−1)/2,is given by a root of unity, q2N = 1, for an integer N . First, for a regular Bethe stateat a root of unity, we show an infinite number of highest weight conditions inductivelyby the algebraic Bethe ansatz method. Here we call a Bethe state regular if it has a setof distinct and finite solutions of the Bethe ansatz equations. In the proof we assume asequence of Bethe roots for generic q whose limit is given by those of the regular Bethestate at the root of unity. Secondly, we prove that a finite-dimensional highest weightrepresentation of the sl2 loop algebra is irreducible. We call a representation highestweight if it is generated by a highest weight vector. We define a Drinfeld polynomialfor a finite-dimensional highest weight representation. We derive the dimensions of thehighest weight representation through the roots of the Drinfeld polynomial. For instance,if the roots of the Drinfeld polynomial P are distinct, the dimensions are given by 2degP .Thirdly, we express the Drinfeld polynomial of a regular XXZ Bethe state at the rootof unity explicitly in terms of the Bethe roots. We also discuss Drinfeld polynomials forBethe ansatz eigenvectors of the inhomogeneous transfer matrix of the six-vertex model. ∗e-mail [email protected]