The primary goal of these lectures is to introduce a beginner to the
finite dimensional representations of Lie groups and Lie algebras.
Since this goal is shared by quite a few other books, we should explain
in this Preface how our approach differs, although the potential reader
can probably see this better by a quick browse through the book.
Representation theory is simple to define: it is the study of the ways
in which a given group may act on vector spaces. It is almost certainly
unique, however, among such clearly delineated subjects, in the breadth
of its interest to mathematicians. This is not surprising: group actions
are ubiquitous in 20th century mathematics, and where the object on
which a group acts is not a vector space, we have learned to replace it
by one that is {e. g. , a cohomology group, tangent space, etc. }. As a
consequence, many mathematicians other than specialists in the field {or
even those who think they might want to be} come in contact with the
subject in various ways. It is for such people that this text is
designed. To put it another way, we intend this as a book for beginners
to learn from and not as a reference. This idea essentially determines
the choice of material covered here. As simple as is the definition of
representation theory given above, it fragments considerably when we try
to get more specific.

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