Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators

TL;DR

This paper explores cyclic representations of quantum affine algebra at roots of unity, establishing their relation to Borel subalgebra representations, and constructs Q-operators satisfying TQ relations for integrable models.

Contribution

It introduces a new connection between cyclic representations at roots of unity and Borel subalgebra tensor products, aiding in Q-operator construction.

Findings

Established relation between cyclic and Borel subalgebra representations at roots of unity.

Constructed Q-operators satisfying TQ relations for 6-vertex and τ₂ models.

Developed exact sequences of representations to facilitate Q-operator construction.

Abstract

We consider the cyclic representations $\Omega_{rs}$ 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 $\Omega_{rs}$ is related to the tensor product $\rho_r\otimes \bar{\rho}_s$ 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 $\tau_2$ models.