On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables basis for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate the basis in which their spectral problem is $separated$, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called "non-fundamental" models we construct two different SoV basis. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second SoV basis for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second SoV basis coincides with the one associated to the Sklyanin's approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solution defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic $Y(gl_{2})$ Yang-Baxter algebra. Our SoV approach also leads to the construction of a $Q$-operator in terms of the fused transfer matrices. Finally, we show that the $Q$-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV basis