commun. Math. Phys. 133,353 368 (1990)zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON Communications inzyxwvutsrq Mathematical Physics © Springer Veriag 1990 Quantum Groups and WZNW M odels A. Alekseev and S. Shatashvili* Leningrad Steklov Mathematical Institute, Fontanka 27, SU 191011, Leningrad, USSR Received December 27, 1989 Abstract. An explanation of the appearance of quantum groups in chiral WZNW models is given. Invariance of the theory under quantum group action is discussed. 1. Introduction Recent developments of various approaches to conformal field theory have led to a deep understanding of the properties of rational conformal field theories (RCFTs). It was shown [1] that monodromy properties of conformal blocks in the WZNW model defines braiding matrices which are closely related to the theory of quantum groups [2].Acategory theoretic point of view on conformal field theory developed by Moore and Seiberg [3] gives evidence that these two subjects, RCFT and the theory of quantum groups are very similar. On the other hand, zyxwvutsrqponmlkjihgfedc E. Witten [4] introduced a universal language both for the conformal field theory and integrable lattice models, where the quantum group plays a key role. But until now, the above observation i.e., coincidence of monodromy properties of conformal blocks in SU(ή ) WZW model with braiding matrices of U q (sl(ή )) has been rather mysterious and in need of a conceptual explanation. In this paper we'll give an explanation of the appearance of JR matrices in conformal field theory and discuss the meaning of invariance of the theory under the quantum group. Our idea is based on the geometric approach to the conformal field theory [5 7]. The main point of our interpretation is that in the chiral WZNW model the dynamics of the element of the loop group, g(x), completely defines the theory and therefore the quantum group must appear at the classical level as a corresponding Poisson Lie group, which acts on the Poisson bracket relations of the loop group element. g(x) itself after quantization contains vertex operators of the theory (see, i.e., [6,8]) and thus quantization of the above Poisson brackets [or, the same, quantum exchange algebra for g(x)] must describe the monodromy * Visitor at the Enrico Fermi Institute, 5640 S. Ellis Ave. Chicago, IL 60637. Supported by NSF PHY 86 57788