Lie groups beyond an introduction / Anthony W. Knapp.
Series Progress in mathematics (Boston, Mass.): v. 140.Editor: Boston : Birkhäuser, c2002Edición: 2nd edDescripción: xviii, 812 p. : il. ; 24 cmISBN: 0817642595 (acidfree paper); 3764342595 (Basel : acid-free paper)Tema(s): Lie groups | Lie algebras | Representations of Lie groupsOtra clasificación: 22-01Preface to the Second Edition xi Preface to the First Edition xiii List of Figures xvi Prerequisites by Chapter xvii Standard Notation xviii INTRODUCTION: CLOSED LINEAR GROUPS [1] 1. Linear Lie Algebra of a Closed Linear Group [1] 2. Exponential of a Matrix [6] 3. Closed Linear Groups [9] 4. Closed Linear Groups as Lie Groups [11] 5. Homomorphisms [16] 6. Problems [20] I. LIE ALGEBRAS AND LIE GROUPS [23] 1. Definitions and Examples [24] 2. Ideals [29] 3. Field Extensions and the Killing Form [33] 4. Semidirect Products of Lie Algebras [38] 5. Solvable Lie Algebras and Lie’s Theorem [40] 6. Nilpotent Lie Algebras and Engel’s Theorem [45] 7. Cartan’s Criterion for Semisimplicity [49] 8. Examples of Semisimple Lie Algebras [56] 9. Representations of sl(2, C) [62] 10. Elementary Theory of Lie Groups [68] 11. Covering Groups [81] 12. Complex Structures [91] 13. Aside on Real-analytic Structures [98] 14. Automorphisms and Derivations [100] 15. Semidirect Products of Lie Groups [102] 16. Nilpotent Lie Groups [106] 17. Classical Semisimple Lie Groups [110] 18. Problems [118] II. COMPLEX SEMISIMPLE LIE ALGEBRAS [123] 1. Classical Root-space Decompositions [124] 2. Existence of Cartan Subalgebras [129] 3. Uniqueness of Cartan Subalgebras [137] 4. Roots [140] 5. Abstract Root Systems [149] 6. Weyl Group [162] 7. Classification of Abstract Cartan Matrices [170] 8. Classification of Nonreduced Abstract Root Systems [184] 9. Serre Relations [186] 10. Isomorphism Theorem [196] 11. Existence Theorem [199] 12. Problems [203] III. UNIVERSAL ENVELOPING ALGEBRA [213] 1. Universal Mapping Property [213] 2. Poincar6-Birkhoff-Witt Theorem [217] 3. Associated Graded Algebra [222] 4. Free Lie Algebras [228] 5. Problems [229] IV. COMPACT LIE GROUPS [233] 1. Examples of Representations [233] 2. Abstract Representation Theory [238] 3. Peter-Weyl Theorem [243] 4. Compact Lie Algebras [248] 5. Centralizers of Tori [251] 6. Analytic Weyl Group [260] 7. Integral Forms [264] 8. Weyl’s Theorem [268] 9. Problems [269] V. FINITE-DIMENSIONAL REPRESENTATIONS [273] 1. Weights [274] 2. Theorem of the Highest Weight [279] 3. Verma Modules [283] 4. Complete Reducibility [290] 5. Harish-Chandra Isomorphism [300] V. FINITE-DIMENSIONAL REPRESENTATIONS Weyl Character Formula [314] Parabolic Subalgebras [325] Application to Compact Lie Groups [333] Problems [339] VI. STRUCTURE THEORY OF SEMISIMPLE GROUPS [347] 1. Existence of a Compact Real Form [348] 2. Cartan Decomposition on the Lie Algebra Level [354] 3. Cartan Decomposition on the Lie Group Level [361] 4. Iwasawa Decomposition [368] 5. Uniqueness Properties of the Iwasawa Decomposition [378] 6. Cartan Subalgebras [384] 7. Cayley Transforms [389] 8. Vogan Diagrams [397] 9. Complexification of a Simple Real Lie Algebra [406] 10. Classification of Simple Real Lie Algebras [408] 11. Restricted Roots in the Classification [422] 12. Problems [426] VII. ADVANCED STRUCTURE THEORY [433] 1. Further Properties of Compact Real Forms [434] 2. Reductive Lie Groups [446] 3. KAK Decomposition [458] 4. Bruhat Decomposition [460] 5. Structure of M [464] 6. Real-rank-one Subgroups [470] 7. Parabolic Subgroups [474] 8. Cartan Subgroups [487] 9. Harish-Chandra Decomposition [499] 10. Problems [514] VIII. INTEGRATION [523] 1. Differential Forms and Measure Zero [523] 2. Haar Measure for Lie Groups [530] 3. Decompositions of Haar Measure [535] 4. Application to Reductive Lie Groups [539] 5. Weyl Integration Formula [547] 6. Problems [552] IX. INDUCED REPRESENTATIONS AND BRANCHING THEOREMS [555] 1. Infinite-dimensional Representations of Compact Groups [556] 2. Induced Representations and Frobenius Reciprocity [563] 3. Classical Branching Theorems [568] 4. Overview of Branching [571] 5. Proofs of Classical Branching Theorems [577] 6. Tensor Products and Littlewood-Richardson Coefficients [596] 7. Littlewood’s Theorems and an Application [602] 8. Problems [609] X. PREHOMOGENEOUS VECTOR SPACES [615] 1. Definitions and Examples [616] 2. Jacobson-Morozov Theorem [620] 3. Vinberg’s Theorem [626] 4. Analysis of Symmetric Tensors [632] 5. Problems [638] APPENDICES A. Tensors, Filiations, and Gradings 1. Tensor Algebra [639] 2. Symmetric Algebra [645] 3. Exterior Algebra [651] 4. Filiations and Gradings [654] 5. Left Noetherian Rings [656] B. Lie’s Third Theorem 1. Levi Decomposition [659] 2. Lie’s Third Theorem [662] 3. Ado’s Theorem [662] 4. Campbell-Baker-Hausdorff Formula [669] C. Data for Simple Lie Algebras 1. Classical Irreducible Reduced Root Systems [683] 2. Exceptional Irreducible Reduced Root Systems [686] 3. Classical Noncompact Simple Real Lie Algebras [693] 4. Exceptional Noncompact Simple Real Lie Algebras [706] Hints for Solutions of Problems [719] Historical Notes [751] References [783] Index of Notation [799] Index [805]
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Incluye referencias bibliográficas (p. 783-798) e índices.
Preface to the Second Edition xi --
Preface to the First Edition xiii --
List of Figures xvi --
Prerequisites by Chapter xvii --
Standard Notation xviii --
INTRODUCTION: CLOSED LINEAR GROUPS [1] --
1. Linear Lie Algebra of a Closed Linear Group [1] --
2. Exponential of a Matrix [6] --
3. Closed Linear Groups [9] --
4. Closed Linear Groups as Lie Groups [11] --
5. Homomorphisms [16] --
6. Problems [20] --
I. LIE ALGEBRAS AND LIE GROUPS [23] --
1. Definitions and Examples [24] --
2. Ideals [29] --
3. Field Extensions and the Killing Form [33] --
4. Semidirect Products of Lie Algebras [38] --
5. Solvable Lie Algebras and Lie’s Theorem [40] --
6. Nilpotent Lie Algebras and Engel’s Theorem [45] --
7. Cartan’s Criterion for Semisimplicity [49] --
8. Examples of Semisimple Lie Algebras [56] --
9. Representations of sl(2, C) [62] --
10. Elementary Theory of Lie Groups [68] --
11. Covering Groups [81] --
12. Complex Structures [91] --
13. Aside on Real-analytic Structures [98] --
14. Automorphisms and Derivations [100] --
15. Semidirect Products of Lie Groups [102] --
16. Nilpotent Lie Groups [106] --
17. Classical Semisimple Lie Groups [110] --
18. Problems [118] --
II. COMPLEX SEMISIMPLE LIE ALGEBRAS [123] --
1. Classical Root-space Decompositions [124] --
2. Existence of Cartan Subalgebras [129] --
3. Uniqueness of Cartan Subalgebras [137] --
4. Roots [140] --
5. Abstract Root Systems [149] --
6. Weyl Group [162] --
7. Classification of Abstract Cartan Matrices [170] --
8. Classification of Nonreduced Abstract Root Systems [184] --
9. Serre Relations [186] --
10. Isomorphism Theorem [196] --
11. Existence Theorem [199] --
12. Problems [203] --
III. UNIVERSAL ENVELOPING ALGEBRA [213] --
1. Universal Mapping Property [213] --
2. Poincar6-Birkhoff-Witt Theorem [217] --
3. Associated Graded Algebra [222] --
4. Free Lie Algebras [228] --
5. Problems [229] --
IV. COMPACT LIE GROUPS [233] --
1. Examples of Representations [233] --
2. Abstract Representation Theory [238] --
3. Peter-Weyl Theorem [243] --
4. Compact Lie Algebras [248] --
5. Centralizers of Tori [251] --
6. Analytic Weyl Group [260] --
7. Integral Forms [264] --
8. Weyl’s Theorem [268] --
9. Problems [269] --
V. FINITE-DIMENSIONAL REPRESENTATIONS [273] --
1. Weights [274] --
2. Theorem of the Highest Weight [279] --
3. Verma Modules [283] --
4. Complete Reducibility [290] --
5. Harish-Chandra Isomorphism [300] --
V. FINITE-DIMENSIONAL REPRESENTATIONS --
Weyl Character Formula [314] --
Parabolic Subalgebras [325] --
Application to Compact Lie Groups [333] --
Problems [339] --
VI. STRUCTURE THEORY OF SEMISIMPLE GROUPS [347] --
1. Existence of a Compact Real Form [348] --
2. Cartan Decomposition on the Lie Algebra Level [354] --
3. Cartan Decomposition on the Lie Group Level [361] --
4. Iwasawa Decomposition [368] --
5. Uniqueness Properties of the Iwasawa Decomposition [378] --
6. Cartan Subalgebras [384] --
7. Cayley Transforms [389] --
8. Vogan Diagrams [397] --
9. Complexification of a Simple Real Lie Algebra [406] --
10. Classification of Simple Real Lie Algebras [408] --
11. Restricted Roots in the Classification [422] --
12. Problems [426] --
VII. ADVANCED STRUCTURE THEORY [433] --
1. Further Properties of Compact Real Forms [434] --
2. Reductive Lie Groups [446] --
3. KAK Decomposition [458] --
4. Bruhat Decomposition [460] --
5. Structure of M [464] --
6. Real-rank-one Subgroups [470] --
7. Parabolic Subgroups [474] --
8. Cartan Subgroups [487] --
9. Harish-Chandra Decomposition [499] --
10. Problems [514] --
VIII. INTEGRATION [523] --
1. Differential Forms and Measure Zero [523] --
2. Haar Measure for Lie Groups [530] --
3. Decompositions of Haar Measure [535] --
4. Application to Reductive Lie Groups [539] --
5. Weyl Integration Formula [547] --
6. Problems [552] --
IX. INDUCED REPRESENTATIONS AND BRANCHING THEOREMS [555] --
1. Infinite-dimensional Representations of Compact Groups [556] --
2. Induced Representations and Frobenius Reciprocity [563] --
3. Classical Branching Theorems [568] --
4. Overview of Branching [571] --
5. Proofs of Classical Branching Theorems [577] --
6. Tensor Products and Littlewood-Richardson Coefficients [596] --
7. Littlewood’s Theorems and an Application [602] --
8. Problems [609] --
X. PREHOMOGENEOUS VECTOR SPACES [615] --
1. Definitions and Examples [616] --
2. Jacobson-Morozov Theorem [620] --
3. Vinberg’s Theorem [626] --
4. Analysis of Symmetric Tensors [632] --
5. Problems [638] --
APPENDICES --
A. Tensors, Filiations, and Gradings --
1. Tensor Algebra [639] --
2. Symmetric Algebra [645] --
3. Exterior Algebra [651] --
4. Filiations and Gradings [654] --
5. Left Noetherian Rings [656] --
B. Lie’s Third Theorem --
1. Levi Decomposition [659] --
2. Lie’s Third Theorem [662] --
3. Ado’s Theorem [662] --
4. Campbell-Baker-Hausdorff Formula [669] --
C. Data for Simple Lie Algebras --
1. Classical Irreducible Reduced Root Systems [683] --
2. Exceptional Irreducible Reduced Root Systems [686] --
3. Classical Noncompact Simple Real Lie Algebras [693] --
4. Exceptional Noncompact Simple Real Lie Algebras [706] --
Hints for Solutions of Problems [719] --
Historical Notes [751] --
References [783] --
Index of Notation [799] --
Index [805] --
MR, MR1920389
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