Algebraic Bethe ansatz for the open XXZ spin chain with non-diagonal boundary terms via $U_q\mathfrak{sl}_2$ symmetry

International audienceWe derive by the traditional Algebraic Bethe Ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [arXiv:hep-th/0304092]. The technical 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 Temperley-Lieb Hamiltonian is realised in a new $U_q\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_q\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules