Linear Lie Groups

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