Cyclic representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel subalgebras at roots of unity and Q-operators

We consider the cyclic representations \Omega_{rs} Ω r s of U_q(\widehat{\mathfrak{sl}}_2) U q ( 𝔰 𝔩 ̂ 2 ) at q^N=1 q N = 1 that depend upon two points r,s r , s in the chiral Potts algebraic curve. We show how \Omega_{rs} Ω r s is related to the tensor product \rho_r\otimes \bar{\rho}_s ρ r ⊗ ρ ‾ s of two representations of the upper Borel subalgebra of U_q(\widehat{\mathfrak{sl}}_2) U q ( 𝔰 𝔩 ̂ 2 ) . This result is analogous to the factorization property of the Verma module of U_q(\widehat{\mathfrak{sl}}_2) U q ( 𝔰 𝔩 ̂ 2 ) at generic- q 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 q N = 1 for both the 6-vertex and \tau_2 τ 2