  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).
