Representations of Quantum Groups

Abstract

The q - deformation U q (G) of the universal enveloping algebras U(G) of complex simple Lie algebras G arose in the study of the algebraic aspects of quantum integrable systems [Fa, KR1, KS, Sl, S2, S3]. (The definition of U q (G)is below in Section 1.) They provide a powerful tool for the solving of the quantum Yang-Baxter equations. For recent reviews we refer to [FaT, FRT1,2, J4, Ta, Vg]. The algebras U q (G) are called also quantum groups [Dl, D2] or quantum universal enveloping algebras [Re, KiR1]. In [S3] for(G)= sl(2,c) and in [Drl, J1, J2, Dr2] in general it was observed that the algebras U q (G) have the structure of a Hopf algebra. This brought additional mathematical interest in this new algebraic structure (see, e.g., [R1, Wo, R2, Ve, L1]). Recently, inspired by the Knizhnik-Zamolodchikov equations [KZ] Drinfeld has developed a theory of formal deformations and introduced a new notion of quasi-Hopf algebras [Dr3,4].