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Representations of Quantum Groups

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Symmetries in Science V

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].

Invited review talk, to appear in the Proceedings of the International Symposium “Symmetries in Science V: Algebraic Structures, their Representations, Realizations and Physical Applications”, Schloss Hofen, Vorarlberg, Austria, 30.7.3.8.1990

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  1. Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Boul. Lenin, 1784, Sofia, Bulgaria

    V. K. Dobrev

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  1. Southern Illinois University, Carbondale, Illinois, USA

    Bruno Gruber

  2. Duke University, Durham, North Carolina, USA

    L. C. Biedenharn

  3. Technical University, Clausthal, Germany

    H. D. Doebner

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Dobrev, V.K. (1991). Representations of Quantum Groups. In: Gruber, B., Biedenharn, L.C., Doebner, H.D. (eds) Symmetries in Science V. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3696-3_5

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