Linear Lie Groups

Name: Linear Lie Groups | Linear Lie Groups
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Academic Press

Published on 21. September 2011

546 pages

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978-0-08-087347-3 (ISBN)

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Front Cover
Linear Lie Groups, Volume 35
Copyright Page
Contents
0. Introduction
0.0 Preface
Chapter 0.1-7. On Notation and Nomenclature
0.1. Logical Symbols
0.2. Set Theory Symbols
0.3. Topological Symbols
0.4. Algebraic Symbols
0.5. Analytic Symbols
0.6. Logograms
0.7. Nomenclature
Chapter 1-5. Preliminaries
1. Complex Extension, Real Restriction, and Waiving
2. The Exponential
3. Some Lie Groups
4. Topological Groups
5 . Differentiable Mappings
Chapter 6-12. The Connection between Local Linear Lie Groups and Lie Algebras
6. Definition of Local and Global Linear Lie Groups
7. The Infinitesimal Algebra of a Local Linear Lie Group
8. The Exponential Presentation
9. Homomorphisms, Automorphisms, and Derivations
10. Expanding Linear Lie Algebras and Their Homomorphisms into Linear Lie Groups and Their Local Homomorphisms
11. Dropping Differentiability Assumptions
12. Subgroups and Subalgebras, Normal Subgroups and ldeals
Chapter 13-19. Solvability and Semisimplicity
13. Solvable Groups and Solvable Lie Algebras
14. Invariants of Linear Lie Groups and Algebras
15. Roots and Rank
16. Important Classes of Complex Lie Algebras
17. Solvable Subalgebras
18. Cleaving
19. Semisimplicity
Chapter 20-27. Dressings and Classification of Semisimple Complex Lie Algebras
20. The First Dressing of Complex Semisimple Lie Algebras
21. The First Weyl Norming and the Second Dressing of Complex Semisimple Lie Algebras
22. G Determined by W*
23. The Second Weyl Norming and the Third Dressing of Complex Semisimple Lie Algebras
24. The Unitary and Standard Restrictions of a Semisimple Lie Algebra
25. G Determined by W++
26. Classification of Semisimple Complex Lie Algebras Up to Isomorphism
27. G2 and F4. The Chevalley Dressing
Chapter 28-38. Topological and Integration Methods
28. Homotopy and Wrapping
29. Fundamental Groups and Wrappings of Topological Groups
30. Compactness Aspects of Semisimple and Abelian Groups
31. The Conjugacy Theorem for Centerfree Unitarily Restricted Semisimple Lie Groups
32. The Fundamental Group of Centerfree Unitarily Restricted Semisimple Groups
33. The Automorphisms of Semisimple Lie Groups
34. Integration in Compact Groups
35. The Conducibility Theorem
36. Orthogonality Relations
37. The Characters of Compact Groups
38. Some Global Properties of Semisimple Linear Lie Groups
Chapter 39-50. The Algebraic Approach to Linear Representations
39. The Associative Envelope of a Lie Algebra
40. The Casimir Tool
41. Weights and Integral Forms
42. Source, Top Weight, and Limitation of a Representation
43. Finite-Dimensional Irreducible Representations
44. The Construction of All Finite-Dimensional Representations
45. The Fundamental Weights
46. The Fundamental Group of Unitarily Restricted Semisimple Lie Groups
47. Weyl's Character and Dimension Formula
48. Algebraic Proof of Weyl's Formulas
49. Clifford Algebras and Spin Representations
50. The Conducibility Theorem (Algebraically Proved) and E. E. Levi's Theorem
Chapter 51-62. Reality in Lie Groups and Algebras and Their Linear Representations
51. Maximally Compact Dressing
52. Classification of Inner Types
53. Classification of Outer Types
54. Further Remarks on Real Classification
55. Contravalence and Virtual Reality of Linear Representations
56. Contravalence of Weights
57. Self-Contravalence
58. Computing e for Simple Lie Algebras
59. Invariant Bilinear and Sesquilinear Forms
60. Minimally Compact Dressing
61. Real Semisimple Linear Lie Groups as Products of Maximal Compact and Solvable Groups
62. The Fundamental Groups of the Real Types
Chapter 63-67. Symmetric Spaces
63. Homogeneous Spaces and Riemannian Manifolds-A Sketch
64. Symmetric Spaces
65. Minimal and Maximal Symmetric Spaces
66. Autometrisms of Symmetric Spaces, Automorphisms of Real Semi- simple Lie Groups
67. Fundamental Groups of Symmetric Spaces
Chapter 68-75. Tits Geometries
68. A List of Fundamental Theorems
69. Proofs of the Statements of Section 68
70. Introduction of Incidence Geometries of Semisimple Lie Groups 70.20. The Special Case of rank 2
71. An Axiomatic Approach to Incidence Geometries of Semisimple Lie Groups
72. Covariants of Pairs of Elements in Incidence Geometries of Semisimple Groups
73. The Classes of Pairs of Elements in an F4-Geometry
74. The Incidence Geometries of Real Semisimple Lie Groups
75. C-Graphs of Incidence Geometries of Simple Semisimple Lie Groups
Chapter 76-77. Betti Numbers of Semisimple Lie Groups and Regular Subalgebras of Semisimple Lie Algebras
76. ad-Nilpotents and Semisimple Subalgebras of Rank 1
77. Killing-Coxeter Tools, Betti Numbers
Appendix
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
Table C. The Positive Rootforms on a Symmetric Basis and Their Altitudes
Table D. Number of Rootforms of Given Positive Altitude a
Table E. Dominant Rootforms
Table F. Fundamental Weights
Table G. Isomorphisms and Equivalences for Low-Rank Semisimple Lie Algebras
Key to Definitions
Author Index
Pure and Applied Mathematics

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Linear Lie Groups

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