Eigenvectors in the Superintegrable Model I: sl₂ Generators

In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic tau_2(t)model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of tau_2(t) in the Q=0 sector is 2^r, with r=(N-1)L/N when the size of the transfer matrix L is a multiple of N. We introduce chiral Potts model operators, different from the more commonly used generators of quantum group U-tilde_q(sl-hat(2)). From these we can form the generators of a loop algebra L(sl(2)). For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra sl(2).