Factorization of the R-matrix and Baxter Q-operators in the $\mathfrak{gl}(1|1)$ spin chain

We study the $\mathfrak{gl}(1|1)$-invariant R-matrix acting on the tensor product of Verma modules. Rather than solving the Yang-Baxter equation directly, we construct the $\mathrm{R}$-matrix from elementary intertwining operators. Our analysis begins with a study of operators intertwining Verma modules. In contrast to the Lie algebra case, a new type of intertwining operator emerges in the super case, related to so-called odd reflections. We then extend our analysis to tensor products and introduce elementary intertwining operators that act by multiplication by a function. Using these intertwining operators, we construct the $\mathrm{R}$-matrix as a product of two commuting operators. A consequence of this local factorization is the factorization of the transfer matrix into a product of Q-operators and the derivation of the TQ-relation.