On the Degenerate Multiplicity of the sl₂ Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Abstract

We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl₂ loop algebra symmetry if the q parameter is given by a root of unity, q₀2N = 1, for an integer N. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight dk±, which leads to evaluation parameters aj. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.