Representations of compact Lie groups / Theodor Bröcker, Tammo tom Dieck.
Series Graduate texts in mathematics ; 98Editor: New York : Springer, 1995Edición: Corr. 2nd printDescripción: x, 313 p. : ill. ; 25 cmISBN: 0387136789 (New York); 3540136789 (Berlin)Tema(s): Lie groups | Representations of Lie groupsOtra clasificación: 22E45 (22-01 57-01)CHAPTER I Lie Groups and Lie Algebras [1] 1. The Concept of a Lie Group and the Classical Examples I 2. Left-Invariant Vector Fields and One-Parameter Groups [11] 3. The Exponential Map [22] 4. Homogeneous Spaces and Quotient Groups [30] 5. Invariant Integration [40] 6. Clifford Algebras and Spinor Groups [54] CHAPTER II Elementary Representation Theory [64] 1. Representations [65] 2. Semisimple Modules [72] 3. Linear Algebra and Representations [74] 4. Characters and Orthogonality Relations [77] 5. Representations of SU(2), SO(3), U(2), and 0(3) [84] 6. Real and Quaternionic Representations [93] 7. The Character Ring and the Representation Ring [102] 8. Representations of Abelian Groups [107] 9. Representations of Lie Algebras [111] 10. The Lie Algebra sl(2,C) [115] CHAPTER III Representative Functions [123] 1. Algebras of Representative Functions [123] 2. Some Analysis on Compact Groups [129] 3. The Theorem of Peter and Weyl I33 4. Applications of the Theorem of Peter and Weyl [136] 5. Generalizations of the Theorem of Peter and Weyl [138] 6. Induced Representations [143] 7. Tannaka-Kreln Duality [146] 8. The Complexification of Compact Lie Groups [151] CHAPTER IV The Maximal Torus of a Compact Lie Group [157] 1. Maximal Tori [157] 2. Consequences of the Conjugation Theorem [164] 3. The Maximal Tori and Weyl Groups of the Classical Groups [169] 4. Cartan Subgroups of Nonconnected Compact Groups [176] CHAPTER V Root Systems [183] 1. The Adjoint Representation and Groups of Rank 1 [183] 2. Roots and Weyl Chambers [189] 3. Root Systems [197] 4. Bases and Weyl Chambers [202] 5. Dynkin Diagrams [209] 6. The Roots of the Classical Groups [216] 7. The Fundamental Group, the Center and the Stiefel Diagram [223] 8. The Structure of the Compact Groups [232] CHAPTER VI Irreducible Characters and Weights [239] 1. The Weyl Character Formula [239] 2. The Dominant Weight and the Structure of the Representation Ring [249] 3. The Multiplicities of the Weights of an Irreducible Representation [257] 4. Representations of Real or Quatemionic Type [261] 5. Representations of the Classical Groups [265] 6. Representations of the Spinor Groups [278] 7. Representations of the Orthogonal Groups [292] Bibliography [299] Symbol Index [305] Subject Index [307]
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Incluye referencias bibliográficas (p. [299]-303) e índices.
CHAPTER I --
Lie Groups and Lie Algebras [1] --
1. The Concept of a Lie Group and the Classical Examples I --
2. Left-Invariant Vector Fields and One-Parameter Groups [11] --
3. The Exponential Map [22] --
4. Homogeneous Spaces and Quotient Groups [30] --
5. Invariant Integration [40] --
6. Clifford Algebras and Spinor Groups [54] --
CHAPTER II --
Elementary Representation Theory [64] --
1. Representations [65] --
2. Semisimple Modules [72] --
3. Linear Algebra and Representations [74] --
4. Characters and Orthogonality Relations [77] --
5. Representations of SU(2), SO(3), U(2), and 0(3) [84] --
6. Real and Quaternionic Representations [93] --
7. The Character Ring and the Representation Ring [102] --
8. Representations of Abelian Groups [107] --
9. Representations of Lie Algebras [111] --
10. The Lie Algebra sl(2,C) [115] --
CHAPTER III --
Representative Functions [123] --
1. Algebras of Representative Functions [123] --
2. Some Analysis on Compact Groups [129] --
3. The Theorem of Peter and Weyl I33 --
4. Applications of the Theorem of Peter and Weyl [136] --
5. Generalizations of the Theorem of Peter and Weyl [138] --
6. Induced Representations [143] --
7. Tannaka-Kreln Duality [146] --
8. The Complexification of Compact Lie Groups [151] --
CHAPTER IV --
The Maximal Torus of a Compact Lie Group [157] --
1. Maximal Tori [157] --
2. Consequences of the Conjugation Theorem [164] --
3. The Maximal Tori and Weyl Groups of the Classical Groups [169] --
4. Cartan Subgroups of Nonconnected Compact Groups [176] --
CHAPTER V --
Root Systems [183] --
1. The Adjoint Representation and Groups of Rank 1 [183] --
2. Roots and Weyl Chambers [189] --
3. Root Systems [197] --
4. Bases and Weyl Chambers [202] --
5. Dynkin Diagrams [209] --
6. The Roots of the Classical Groups [216] --
7. The Fundamental Group, the Center and the Stiefel Diagram [223] --
8. The Structure of the Compact Groups [232] --
CHAPTER VI --
Irreducible Characters and Weights [239] --
1. The Weyl Character Formula [239] --
2. The Dominant Weight and the Structure of the Representation Ring [249] --
3. The Multiplicities of the Weights of an Irreducible Representation [257] --
4. Representations of Real or Quatemionic Type [261] --
5. Representations of the Classical Groups [265] --
6. Representations of the Spinor Groups [278] --
7. Representations of the Orthogonal Groups [292] --
Bibliography [299] --
Symbol Index [305] --
Subject Index [307] --
MR, MR1410059
"This book is an introduction to the representation theory of compact Lie groups, following Hermann Weyl's original approach. Although the authors discuss all aspects of finite-dimensional Lie theory, the emphasis throughout the book is on the groups themselves. The presentation is consequently more geometric and analytic than algebraic in nature. The central results, culminating the Weyl character formula, are reached directly and quickly, and they appear in forms suitable for applications to physics and geometry.This book is a good reference and a source of explicit computations, for physicists and mathematicians. Each section is supplemented by a wide range of exercices, and geometric ideas are illustrated with the help of 24 figures."--Contratapa.
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