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

We consider the cyclic representations ${\mathrm{\Omega }}_{rs}$ of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how ${\mathrm{\Omega }}_{rs}$ is related to the tensor product ${\rho }_r\otimes {\bar{\rho }}_s$ of two representations of the upper Borel subalgebra of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ 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.