Quasi-Hopf twistors for elliptic quantum groups

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U_q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U_q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. This construction entails that, for generic values of the deformation parameters, representation theory for U_q(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra A_{q,p}(^sl_2).