Representation Theory: A First Course

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Springer Science & Business Media, 01.12.2013 - 551 Seiten
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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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Inhalt

Contents
1
Characters
12
of topics that are not logically essential to the rest of the book and that
26
Character Formula
48
Representations of QI and GL2F
63
Weyls Construction
75
Lie Groups and Lie Algebras
89
Lie Algebras and Lie Groups
104
Orthogonal Lie Algebras
267
Spin Representations of 300
299
Lie Theory 317
316
g2 and Other Exceptional Lie Algebras
339
Complex Lie Groups Characters
366
Weyl Character Formula
399
More Character Formulas
415
Real Lie Algebras and Lie Groups
430

Initial Classification of Lie Algebras
121
Lie Algebras in Dimensions One Two and Three
133
Representations of sl2C
146
Representations of sla C Part I
161
Mainly Lots of Examples
175
The Classical Lie Algebras and Their Representations 195
194
3I4C and slC
211
Symplectic Lie Algebras
238
pe C and sp2C
253
Appendices
450
On Semisimplicity
478
Cartan Subalgebras
487
E Ados and Levis Theorems
499
Hints Answers and References
516
Bibliography
536
Index of Symbols
543
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