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Quasi-Hopf twistors for elliptic quantum groups. (English) Zbl 0977.17012

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 O. Foda, K. Iohara, M. Jimbo, R. Kedem, T. Miwa and H. Yan, Lett. Math. Phys. 32, 259–268 (1994; Zbl 0821.17011)] and by G. Felder [Proc. Int. Congress Math. Phys., Paris 1994, 211–218 (1995; Zbl 0998.17015)] respectively. Later 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.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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