On $τ^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$
Abstract
Description
We identify the precise relationship between the five-parameter $τ^{(2)}$-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 $τ^{(2)}$-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 $τ^{(2)}$-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 $τ^{(2)}$-model.
Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal version
Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal version