We identify the precise relationship between the five-parameter $\tau^{ (2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state $\tau^{ (2)}$-model can be regarded as the XXZ chain of $U_{ \sf q} (sl_2)$ cyclic representations with ${ \sf q}^N=1$. The symmetry algebra of the $\tau^{ (2)}$-model is described by the quantum affine algebra $U_{ \sf q} (\hat{ sl}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for $q^{ 2N}=1$ is equivalent to two copies of the same $N$-state $\tau^{ (2)}$-model.
Categories: cond-mat.stat-mech hep-th math-ph math.MP math.QA nlin.SI