This is an introduction to the foundations of quantum group theory. Quantum groups or Hopf algebras are an exciting new generalisation of ordinary groups. They have a rich mathematical structure and numerous roles in situations where ordinary groups are not adequate. The goal in this volume is to set out this mathematical structure by developing the basic properties of quantum groups as objects in their own right; what quantum groups are conceptually and how to work with them. We will also give some idea of the meaning of quantum groups for physics. On the other hand, just as ordinary groups have all sorts of applications in physics, not one specific application but many, in the same way one finds that quantum groups have a wide variety of probably unrelated applications. This diversity is one of the themes in the volume and is a good reason to focus on quantum groups as mathematical objects.

This book is not a survey; many of the most interesting recent results in representation theory, applications in conformal field theory and lowdimensional topology, etc., are not discussed in any detail. In this sense, there is less material here than in my lecture notes [1]. In place of this fashionable material, I have developed the pedagogical side of [1], giving now more details of proofs and solutions to exercises, and in general concentrating more on that part of the theory of quantum groups that can be considered as firmly established.