The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(s{l}_2)$

We demonstrate that the transfer matrix of the inhomogeneous $N$-state chiral Potts model with two vertical superintegrable rapidities serves as the $Q$-operator of XXZ chain model for a cyclic representation of $U_{\sf q}(sl_2)$with$N$throot-of-unity${\sf q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sf q}(sl_2)$-representationismappedontotheproblemofstudying$Q$-operator of some special one-parameter family of generalized ${\tau }^{(2)}$-models. In particular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\mathsf{q}}^N=1$ and the homogeneous $N$-state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter’s method developed for producing $Q_{72}$-operator of the root-of-unity eight-vertex model, we construct the $Q_R,Q_L$- and $Q$-operators of a superintegrable ${\tau }^{(2)}$-model, then identify them with transfer matrices of the $N$-state chiral Potts model for a positive integer $N$. We thus obtain a new method of producing the superintegrable $N$-state chiral Potts transfer matrix from the ${\tau }^{(2)}$-model by constructing its $Q$-operator.