Quantum Groups

{( Eh bien, Monsieur, que pensez-vous des x et des y ?» Je lui ai repondu : {( C'est bas de plafond. » V. Hugo [Hug51] The term "quantum groups" was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley (1986). It stands for certain special Hopf algebras which are nontrivial deformations of the enveloping Hopf algebras of semisimple Lie algebras or of the algebras of regular functions on the corresponding algebraic groups. As was soon ob served, quantum groups have close connections with varied, a priori remote, areas of mathematics and physics. The aim of this book is to provide an introduction to the algebra behind the words "quantum groups" with emphasis on the fascinating and spec tacular connections with low-dimensional topology. Despite the complexity of the subject, we have tried to make this exposition accessible to a large audience. We assume a standard knowledge of linear algebra and some rudiments of topology (and of the theory of linear differential equations as far as Chapter XIX is concerned).