Linear Lie Groups (eBook)

Front Cover 1
Linear Lie Groups, Volume 35 4
Copyright Page 5
Contents 6
0. Introduction 16
0.0 Preface 16
Chapter 0.1-7. On Notation and Nomenclature 18
0.1. Logical Symbols 18
0.2. Set Theory Symbols 18
0.3. Topological Symbols 19
0.4. Algebraic Symbols 20
0.5. Analytic Symbols 21
0.6. Logograms 21
0.7. Nomenclature 21
Chapter 1-5. Preliminaries 26
1. Complex Extension, Real Restriction, and Waiving 26
2. The Exponential 28
3. Some Lie Groups 35
4. Topological Groups 36
5 . Differentiable Mappings 45
Chapter 6-12. The Connection between Local Linear Lie Groups and Lie Algebras 48
6. Definition of Local and Global Linear Lie Groups 48
7. The Infinitesimal Algebra of a Local Linear Lie Group 49
8. The Exponential Presentation 56
9. Homomorphisms, Automorphisms, and Derivations 60
10. Expanding Linear Lie Algebras and Their Homomorphisms into Linear Lie Groups and Their Local Homomorphisms 67
11. Dropping Differentiability Assumptions 71
12. Subgroups and Subalgebras, Normal Subgroups and ldeals 77
Chapter 13-19. Solvability and Semisimplicity 88
13. Solvable Groups and Solvable Lie Algebras 88
14. Invariants of Linear Lie Groups and Algebras 95
15. Roots and Rank 98
16. Important Classes of Complex Lie Algebras 100
17. Solvable Subalgebras 106
18. Cleaving 113
19. Semisimplicity 117
Chapter 20-27. Dressings and Classification of Semisimple Complex Lie Algebras 124
20. The First Dressing of Complex Semisimple Lie Algebras 124
21. The First Weyl Norming and the Second Dressing of Complex Semisimple Lie Algebras 132
22. G Determined by W* 136
23. The Second Weyl Norming and the Third Dressing of Complex Semisimple Lie Algebras 140
24. The Unitary and Standard Restrictions of a Semisimple Lie Algebra 142
25. G Determined by W++ 147
26. Classification of Semisimple Complex Lie Algebras Up to Isomorphism 156
27. G2 and F4. The Chevalley Dressing 162
Chapter 28-38. Topological and Integration Methods 167
28. Homotopy and Wrapping 167
29. Fundamental Groups and Wrappings of Topological Groups 175
30. Compactness Aspects of Semisimple and Abelian Groups 177
31. The Conjugacy Theorem for Centerfree Unitarily Restricted Semisimple Lie Groups 179
32. The Fundamental Group of Centerfree Unitarily Restricted Semisimple Groups 185
33. The Automorphisms of Semisimple Lie Groups 191
34. Integration in Compact Groups 204
35. The Conducibility Theorem 209
36. Orthogonality Relations 212
37. The Characters of Compact Groups 220
38. Some Global Properties of Semisimple Linear Lie Groups 229
Chapter 39-50. The Algebraic Approach to Linear Representations 232
39. The Associative Envelope of a Lie Algebra 232
40. The Casimir Tool 236
41. Weights and Integral Forms 238
42. Source, Top Weight, and Limitation of a Representation 242
43. Finite-Dimensional Irreducible Representations 246
44. The Construction of All Finite-Dimensional Representations 250
45. The Fundamental Weights 254
46. The Fundamental Group of Unitarily Restricted Semisimple Lie Groups 259
47. Weyl’s Character and Dimension Formula 263
48. Algebraic Proof of Weyl’s Formulas 268
49. Clifford Algebras and Spin Representations 273
50. The Conducibility Theorem (Algebraically Proved) and E. E. Levi’s Theorem 281
Chapter 51-62. Reality in Lie Groups and Algebras and Their Linear Representations 290
51. Maximally Compact Dressing 290
52. Classification of Inner Types 306
53. Classification of Outer Types 312
54. Further Remarks on Real Classification 316
55. Contravalence and Virtual Reality of Linear Representations 320
56. Contravalence of Weights 323
57. Self-Contravalence 325
58. Computing e for Simple Lie Algebras 331
59. Invariant Bilinear and Sesquilinear Forms 335
60. Minimally Compact Dressing 345
61. Real Semisimple Linear Lie Groups as Products of Maximal Compact and Solvable Groups 360
62. The Fundamental Groups of the Real Types 362
Chapter 63-67. Symmetric Spaces 374
63. Homogeneous Spaces and Riemannian Manifolds—A Sketch 374
64. Symmetric Spaces 387
65. Minimal and Maximal Symmetric Spaces 404
66. Autometrisms of Symmetric Spaces, Automorphisms of Real Semi- simple Lie Groups 407
67. Fundamental Groups of Symmetric Spaces 416
Chapter 68-75. Tits Geometries 420
68. A List of Fundamental Theorems 420
69. Proofs of the Statements of Section 68 425
70. Introduction of Incidence Geometries of Semisimple Lie Groups 70.20. The Special Case of rank 2 439
71. An Axiomatic Approach to Incidence Geometries of Semisimple Lie Groups 452
72. Covariants of Pairs of Elements in Incidence Geometries of Semisimple Groups 461
73. The Classes of Pairs of Elements in an F4-Geometry 464
74. The Incidence Geometries of Real Semisimple Lie Groups 504
75. C-Graphs of Incidence Geometries of Simple Semisimple Lie Groups 510
Chapter 76-77. Betti Numbers of Semisimple Lie Groups and Regular Subalgebras of Semisimple Lie Algebras 522
76. ad-Nilpotents and Semisimple Subalgebras of Rank 1 522
77. Killing-Coxeter Tools, Betti Numbers 526
Appendix 552
Table A. The Graph and Dimension of G e Alg Lie Com SSS The Length Square of the Shortest Nonzero Rootform
Table B. The Positive Rootforms on a Natural Basis 553
Table C. The Positive Rootforms on a Symmetric Basis and Their Altitudes 557
Table D. Number of Rootforms of Given Positive Altitude a 559
Table E. Dominant Rootforms 560
Table F. Fundamental Weights 561
Table G. Isomorphisms and Equivalences for Low-Rank Semisimple Lie Algebras 563
Key to Definitions 564
Author Index 572
Pure and Applied Mathematics 573