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arXiv · 2412.14811

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

Robert Weston
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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.

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Robert Weston. 2024-12-19. Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators. https://doi.org/10.21468/scipostphyscore.9.1.016

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