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

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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 sl₂ loop algebra. Here q is related to the XXZ anisotropic coupling Δ by Δ=(q+q⁻¹)/2, and it is given by a root of unity, q²ⁿ=1, for a positive integer N. We show that regular XXZ Bethe states are annihilated by the generators xₖ⁺'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, hₖ's. Here the loop algebra U(L(sl₂)) is generated by xₖ± and hₖ for k ∈ Z, which are the classical analogues of the Drinfeld generators of the quantum loop algebra Uq(L(sl₂)). A representation of U(L(sl₂)) is called highest weight if it is generated by a vector Ω which is annihilated by the generators xₖ⁺'s and such that Ω is an eigenvector of the hₖ's. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of U(L(sl₂)).