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

We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the $sl_2$ loop algebra. Here $q$ is related to the XXZ anisotropic coupling $\Delta$ by $\Delta=(q+q^{-1})/2$, and it is given by a root of unity, $q^{2N}=1$, for a positive integer $N$. We show that regular XXZ Bethe states are annihilated by the generators ${\bar x}_k^{+}$'s, for any $N$. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra, ${\bar h}_k$'s. Here the loop algebra $U(L(sl_2))$ is generated by ${\bar x}_k^{\pm}$ and ${\bar h}_k$ for $k \in {\bf Z}$, which are the classical analogues of the Drinfeld generators of the quantum loop algebra $U_q(L(sl_2))$. A representation of $U(L(sl_2))$ is called highest weight if it is generated by a vector $\Omega$ which is annihilated by the generators ${\bar x}_k^{+}$'s and such that $\Omega$ is an eigenvector of the ${\bar h}_k$'s. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of $U(L(sl_2))$.