On ${\tau }^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(s{l}_2)$

Abstract

We identify the precise relationship between the five-parameter ${\tau }^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q(s{l}_2)$-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 $U_q(s{l}_2)$. When $N$ is odd, the $N$-state ${\tau }^{(2)}$-model can be regarded as the XXZ chain of $U_{\mathsf{q}}(s{l}_2)$ cyclic representations with ${\mathsf{q}}^N=1$. The symmetry algebra of the ${\tau }^{(2)}$-model is described by the quantum affine algebra $U_{\mathsf{q}}({\widehat{sl}}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q(s{l}_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state ${\tau }^{(2)}$-model.

Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal version

References in corpus (3)

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