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.
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