Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators
Robert Weston
TL;DR
This work develops a systematic root-of-unity framework for $U_q(\hat{\mathfrak{sl}}_2)$ by constructing finite-dimensional cyclic representations $\Omega_{rs}$ and upper-Borel representations $\rho_r,\bar{\rho}_r,\varphi_c$ of $U_q(\mathfrak{b}_+)$. It proves a key factorization ${\mathcal{O}}(\chi): \Omega_{rs}\otimes \varphi_{c_0} \to \rho_r\otimes \bar{\rho}_s$ and establishes short exact sequences that encode fusion, together with a compatible L-operator factorization. These structural results yield Q-operator constructions for the $6$-vertex model and the $\tau_2$/Chiral Potts setting, satisfying standard TQ relations (up to gauge) and linking to the CP half-monodromy picture of Baxter’s framework. The finite-dimensional, root-of-unity setting opens avenues for higher-rank generalizations and open-system Q-operators, highlighting a coherent algebraic path from Borel representations to transfer matrices and QQ-relations. Overall, the paper clarifies how cyclic $U_q(\mathfrak{b}_+)$ representations underpin Q-operators at roots of unity and how they connect to the chiral Potts/Baxter framework.
Abstract
We consider the cyclic representations $Ω_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $Ω_{rs}$ is related to the tensor product $ρ_r\otimes \barρ_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $τ_2$ models.