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Representation Theory

A First Course

By: Joe Harris, William Fulton

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Paperback | 1 January 1991 | Edition Number 3

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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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You Can Find This Book In

Non-Fiction Mathematics Groups & Group Theory Algebra Mathematical Foundation Set Theory Calculus & Mathematical Analysis Functional Analysis & Transforms Topology

Finite Groups
Representations of Finite Groups
Characters
Examples; Induced Representations; Groups Algebras; Real Representations
Representations of Ed: Young Diagrams and Frobenius's Character Formula
Representation of Ud and GL2 and (Fq)
Weyl's Construction
Lie Groups and Lie Algebras
Lie Groups
Lie Algebras and Lie Groups
Initial Classification of Lie Algebras
Lie Algebras in Dimensions One, Two, and Three
Representation of sI2 C
Representation of sI3 C, Part I
Representation of sI3 C, II: Mainly Lots of Examples
The Classical Lie Algebras and Their Representations
The General Set-up: Analyzing the Structure and Representations of an Arbitrary Semi simple Lie Algebra
SI4 C and sIn C
Symplectic Lie Algebras
Sp6C and sp2n C
Orthogonal Lie Algebras
So6 C, so7 C, and som C
Spin Representations of so m C
Lie Theory
The Classification of Complex Simple Lie Algebras
G2 and Other Exceptional Lie Algebras
Complex Lie Groups; Characters
Weyl Character Formula
More Character Formulas
Real Lie Algebras and Lie Groups
Appendices A - F
Hints
Answers
References
Bibliography
Index of Symbols
Index
Table of Contents provided by Publisher. All Rights Reserved.

Representation Theory

A First Course

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