00292 - An Elementary Introduction to Groups and Representations.pdf

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An Elementary Introduction to Groups and
Representations
Brian C. Hall
Author address:
University of Notre Dame, Department of Mathematics, Notre
Dame IN 46556 USA
E-mail address : bhall@nd.edu
Contents
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1. PREFACE
v
1. Preface
) : Although these are
often called simply \matrix groups," my terminology emphasizes that every matrix
group is a Lie group.
This approach to the subject allows me to get started quickly on Lie group the-
ory proper, with a minimum of prerequisites. Since most of the interesting examples
of Lie groups are matrix Lie groups, there is not too much loss of generality. Fur-
thermore, the proofs of the main results are ultimately similar to standard proofs
in the general setting, but with less preparation.
Of course, there is a price to be paid and certain constructions (e.g. covering
groups) that are easy in the Lie group setting are problematic in the matrix group
setting. (Indeed the universal cover of a matrix Lie group need not be a matrix
Lie group.) On the other hand, the matrix approach suces for a rst course.
Anyone planning to do research in Lie group theory certainly needs to learn the
manifold approach, but even for such a person it might be helpful to start with a
more concrete approach. And for those in other elds who simply want to learn
the basics of Lie group theory, this approach allows them to do so quickly.
These notes also use an atypical approach to the theory of semisimple Lie
algebras, namely one that starts with a detailed calculation of the representations
of sl (3;
C
). My own experience was that the theory of Cartan subalgebras, roots,
Weyl group, etc., was pretty dicult to absorb all at once. I have tried, then, to
motivate these constructions by showing how they are used in the representation
theory of the simplest representative Lie algebra. (I also work out the case of
sl (2;
C
) ; but this case does not adequately illustrate the general theory.)
In the interests of making the notes accessible to as wide an audience as possible,
I have included a very brief introduction to abstract groups, given in Chapter 1.
In fact, not much of abstract group theory is needed, so the quick treatment I give
should be sucient for those who have not seen this material before.
I am grateful to many who have made corrections, large and small, to the notes,
including especially Tom Goebeler, Ruth Gornet, and Erdinch Tatar.
These notes are the outgrowth of a graduate course on Lie groups I taught
at the University of Virginia in 1994. In trying to nd a text for the course I
discovered that books on Lie groups either presuppose a knowledge of dierentiable
manifolds or provide a mini-course on them at the beginning. Since my students
did not have the necessary background on manifolds, I faced a dilemma: either use
manifold techniques that my students were not familiar with, or else spend much
of the course teaching those techniques instead of teaching Lie theory. To resolve
this dilemma I chose to write my own notes using the notion of a matrix Lie group .
A matrix Lie group is simply a closed subgroup of GL ( n ;
C

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