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We demonstrate that the transfer matrix of the inhomogeneous $N$-state chiralPotts model with two vertical superintegrable rapidities serves as the$Q$-operator of XXZ chain model for a cyclic representation of $U_{\sfq}(sl_2)$ with $N$th root-of-unity ${\sf q}$ and representation-parameter forodd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sfq}(sl_2)$-representation is mapped onto the problem of studying $Q$-operator ofsome special one-parameter family of generalized $\tau^{(2)}$-models. Inparticular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\sf q}^N=1$ and thehomogeneous $N$-state chiral Potts model at a specific superintegrable pointare 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, thenidentify them with transfer matrices of the $N$-state chiral Potts model for apositive integer $N$. We thus obtain a new method of producing thesuperintegrable $N$-state chiral Potts transfer matrix from the$\tau^{(2)}$-model by constructing its $Q$-operator.

The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(sl_2)$