2 INTRODUCTION

That course was directed to students who had just been introduced to complex

semisimple Lie algebras along the lines of the textbook by Humphreys [H]. The goal

of the course (and of the notes) was to make them acquainted with basic parts of the

theory of quantum groups. Later I added to the notes one important application

(the canonical or crystal bases) for which there had not been enough time in the

actual course.

At the time of that course there was only one book on the subject available to

us, the one by Lusztig. I felt that my students were not adequately prepared for

his text, and I wrote my notes with the hope to provide this preparation. In the

meantime a few other books on the subject have appeared (by Chari & Pressley,

by Joseph, and by Kassel). Except for the one by Kassel these books appear to be

directed mainly to the research mathematician, leading to the current frontier of

the research in these areas. In contrast, this book is meant for the student learning

the subject for the first time. So I do not work in the greatest generality possible. I

assume that the reader is familiar with the main facts about semisimple Lie algebras

(as in [H]), and I try to emphasize similarities and differences with that classical

theory.

The book by Kassel is also directed to non-experts or not-yet-experts. However,

it goes into a direction quite different from the one here. It discusses many beautiful

applications of quantum groups, in particular to knot theory. At the same time it

restricts itself mainly to the case of g = s^ (with a few hints made to the general

case).

All these other books can provide useful supplemental or continuing reading.

So will the survey article by De Concini and Procesi in Springer Lecture Notes

1565.

Let me now briefly describe how this book is organized. It begins with a short

preliminary Chapter (0) collecting basic properties of Gaussian binomial coeffi-

cients. Then there are three Chapters (1-3) dealing with U = Uq(g) in the special

case Q = s{2. We see here in an example many of the features to be discussed later

on in general (bases, simple modules, Hopf algebra structure). Besides, the results

in this special case will be applied in the treatment of the general case.

When we classify simple Uq (sI2)-modules, we observe a clear division into two

cases: If q is not a root of unity, then the representation theory of t/^sfe) looks like

that of the Lie algebra 3X2 in characteristic 0. If q is a root of unity, then it looks

like the representation theory of 0(2 in prime characteristic. Both facts generalize

to arbitrary g. While we shall deal with the non-root-of-unity case in general in

Chapter 5, we do not treat the second case in this book (for arbitrary g). One can

find an introduction to that topic in [De Concini & Procesi].

In Chapter 4 we state the definition of Uq(#) for general g and show that there

is a "triangular decomposition" of Uq (g). We deal here (and throughout the book)

only with finite dimensional g and do not discuss possible generalisations to Kac-

Moody algebras.

We turn then to the representation theory of Uq(g). Prom now on we assume

that q is not a root of unity. We classify (in Chapter 5) the simple finite dimensional

Uq (g)-modules and show that all finite dimensional C/q(g)-modules are semisimple

(unless we are in characteristic 2). In Chapter 5A we describe several examples of

simple modules explicitly.

For some additional results in Chapter 5 we assume that k has characteristic 0

and that q is transcendental over Q. This restriction turns out to be unnecessary