Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via Uqsl2 Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equa-tions for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The tech-nical difficulties due to the breaking of U(1) symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temp erley-Lieb Hamiltonian is realised in a new Uqsl2-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur- Weyl duality between Uqsl2 and the two-boundary Temp erley-Lieb algebra. In this frame-work, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.