Structure of the two-boundary X X Z model with non-diagonal boundary terms

We study the integrable XXZ model with general non-diagonal boundary terms at both ends. The Hamiltonian is considered in terms of a two-boundary extension of the Temperley-Lieb algebra. We use a basis that diagonalizes a conserved charge in the one-boundary case. The action of the second boundary generator on this space is computed. For the L-site chain and generic values of the parameters we have an irreducible space of dimension 2(L). However, at certain critical points there exists a smaller irreducible subspace that is invariant under the action of all the bulk and boundary generators. These are precisely the points at which Bethe ansatz equations have been formulated. We compute the dimension of the invariant subspace at each critical point and show that it agrees with the splitting of eigenvalues, found numerically, between the two Bethe ansatz equations.