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
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