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We derive by the traditional algebraic Bethe ansatz method the Betheequations for the general open XXZ spin chain with non-diagonal boundary termsunder the Nepomechie constraint [J. Phys. A 37 (2004), 433-440,arXiv:hep-th/0304092]. The technical difficulties due to the breaking of$\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome byan algebraic construction where the two-boundary Temperley-Lieb Hamiltonian isrealised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chaininvolving infinite-dimensional Verma modules on the edges [J. High Energy Phys.2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of thetwo Hamiltonians is established by proving Schur-Weyl duality between$U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra.In this framework, the Nepomechie condition turns out to have a simplealgebraic interpretation in terms of quantum group fusion rules.

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry