INTRODUCTION

3

as we shall see in Chapter 8. But it requires hard work to remove the restriction,

and so I give the easy proof in the special case early on.

In Chapter 6 we determine the center of Uq(g) and get an analogue of the

classical Harish-Chandra theorem on the center of U(g). The proof involves a

special feature of Uq(g): a non-degenerate bilinear form on Uq(g) that (in some

sense) plays the role of the Killing form on g.

In Chapter 8 we construct a basis of Uq(g) similar to the PBW basis of 17(g).

This basis behaves nicely with respect to the bilinear form just mentioned; that is

proved in Chapter 8A.

The earlier Chapter 7 discusses the connection between Uq(g) - the deformation

of U(g) — and kq[G\, the deformation of the algebra of regular functions on G.

Furthermore we look at R-matrices. These are (in our context) isomorphisms

M (g M' ^ M' ® M for certain finite dimensional i7g(g)-modules M and M'.

They lead to solutions of the so-called quantum Yang-Baxter equations and were

the first reason for introducing quantum groups.

The last three Chapters (9-11) deal with the crystal or canonical bases. Here we

have an example where the investigation of quantum groups has led to new results

on the original Lie algebra g: Take a triangular decomposition g = n~ 0 J) © n

+

.

Then there is a basis B for the enveloping algebra U(n~) of n~ such that for each

simple finite dimensional g-module V with highest weight vector v the uv with

u G B and uv ^ 0 are a basis of V. This result is proved by finding first such a

basis for the subalgebra U~ of Uq(g) that is analogous to U(n~).

The theory of quantum groups involves occasionally long computations. I have

moved several of them (as well as a few more straightforward calculations) to the

end of the corresponding Chapter. The reader may want to do some of them as

exercises without looking first at these appendices.

I have not tried to attribute credit for all the results in this book. The list of

references has been restricted to the sources that I have used, to more advanced

books in the area (where one can find a more extensive biliography), and to a few

books that are quoted in the text.

I want to thank everyone who has pointed out misprints and inaccuracies in

the first version of my notes, in particular Patrick Brewer, George McNinch, Jens

Gunner Jensen and Jim Humphreys.