Quasi-Hopf twistors for elliptic quantum groups
The quantum dynamical Yang-Baxter equation admits two types of elliptic solutions. One type of solutions corresponds to face-type integrable models, while the other to vertex-type models. This has led to the introduction of two types of elliptic quantum groups by \textit{O. Foda, K. Iohara, M. Jimbo, R. Kedem, T. Miwa} and \textit{H. Yan}, Lett. Math. Phys. 32, 259--268 (1994; Zbl 0821.17011)] and by \textit{G. Felder} [Proc. Int. Congress Math. Phys., Paris 1994, 211--218 (1995; Zbl 0998.17015)] respectively. Later \textit{C. Fronsdal} [Publ. Res. Inst. Math. Sci. 33, 91--149 (1997; Zbl 0899.16021), Lett. Math. Phys. 40, 117--134 (1997; Zbl 0882.17006)] has shown that both types of elliptic quantum groups are in fact quasi-Hopf algebras obtained from the standard quantum affine algebra \(U_q(g)\) by twisting. In this paper the authors present an explicit formula for twists in the form of an infinite product of the universal \(R\)-matrix of \(U_q(g)\). The presented construction shows that, for generic values of deformation parameters, various representation-theoretic properties of elliptic quantum groups can be obtained from the corresponding properties of the standard quantum groups.
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