Representation of the boundary elliptic quantum group BEτ, η( sl2) and the Bethe ansatz

We extend the Gervais-Neveu-Felderequation (GNFE) to the boundary case. Using the boundary GNFE, we give a definition of the boundary elliptic quantum group BE_{τ, η}( sl₂) The evaluation representation of the boundary elliptic quantum group BE_{τ, η}( sl₂) is presented. The Bethe ansatz for the boundary quantum group BE_{τ, η}( sl₂) is obtained. In a special case, the obtained Bethe ansatz is similar to the Bethe ansatz of the eight-vertex and the higher spin eight-vertex models with open boundary conditions.