Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995
(Russian translation: Phasis, Moscow, 1999)

Christian Kassel

Quantum Groups

Mathematics Subject Classification (1991): Primary 17B37, 18D10, 57M25, 81R50; Secondary 16W30, 17B20, 17B35, 18D99, 20F36

Abstract. This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In Part Three we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. Part Four is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the contruction of Kontsevich's universal knot invariant.

A list of errata (updated on 4 September 2001) is available:

Russian translation: "Kvantovye ", (translator: I. A. Dynnikov, editor: V. M. Buchstaber), Biblioteka Mathematika, vol. 5, Phasis, Moscow, 1999 (20 chapitres, 85 figures, 663 p. + xxi). ISBN 5-7036-0052-9.