Quantum Group Theory in ${\tau }^{(2)}$-model, Duality of ${\tau }^{(2)}$-model and XXZ-model with Cyclic ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-representation for ${\mathbf{q}}^{\mathbf{n}}=1$, and Chiral Potts Model

Abstract

We identify the quantum group ${\Large\textsl{U}}_\textsl{w}(sl_2)$ in the $L$-operator of ${\tau }^{(2)}$-model for a generic $\textsl{w}$ as a subalgebra of $U_{\mathsf{q}}(s{l}_2)$ with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with $q^{\mathbf{n}}={\omega }^N=1$, the eigenvalues and eigenvectors of XXZ-model with the $U_q(s{l}_2)$-cyclic representation are determined by the ${\tau }^{(2)}$-model with the induced ${\Large\textsl{U}}_ω(sl_2)$-cyclic representation, which is decomposed as a finite sum of ${\tau }^{(2)}$-models in non-superintegrable inhomogeneous $N$-state chiral Potts model. Through the theory of chiral Potts model, the $Q$-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the $\mathbf{n}=2N$, e.g. $N$ even, case. We also establish the duality of ${\tau }^{(2)}$-models related to cyclic representations of $U_q(s{l}_2)$, analogous to the ${\tau }^{(2)}$-duality in chiral Potts model; and identify the model dual to the XXZ model with $U_q(s{l}_2)$-cyclic representation.

arXiv commentLatex 28 pages