Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926

Front Cover
Springer Science & Business Media, Jul 19, 2000 - Mathematics - 566 pages
This book is both more and less than a history of the theory of Lie groups during the period 1869-1926. No attempt has been made to provide an exhaustive treatment of all aspects of the theory. Instead, I have focused upon its origins and upon the subsequent development of its structural as pects, particularly the structure and representation of semisimple groups. In dealing with this more limited subject matter, considerable emphasis has been placed upon the motivation behind the mathematics. This has meant paying close attention to the historical context: the mathematical or physical considerations that motivate or inform the work of a particular mathematician as well as the disciplinary ideals of a mathematical school that encourage research in certain directions. As a result, readers will ob tain in the ensuing pages glimpses of and, I hope, the flavor of many areas of nineteenth and early twentieth century geometry, algebra, and analysis. They will also encounter many of the mathematicians of the period, includ ing quite a few not directly connected with Lie groups, and will become acquainted with some of the major mathematical schools. In this sense, the book is more than a history of the theory of Lie groups. It provides a different perspective on the history of mathematics between, roughly, 1869 and 1926. Hence the subtitle.
 

Contents

The Geometrical Origins of Lies Theory
1
11 Tetrahedral Line Complexes
2
12 WCurves and WSurfaces
10
13 Lies Idee Fixe
20
14 The Sphere Mapping
26
15 The Erlanger Programm
34
Jacobi and the Analytical Origins of Lies Theory
43
21 Jacobis Two Methods
44
73 Gino Fano
251
74 Cayleys Counting Problem
260
75 Kowalewskis Theory of Weights
265
Cartans Trilogy 191314
277
81 Research Priorities 18931909
278
82 Another Application of Secondary Roots
287
83 Continuous Groups and Geometry
290
84 The Memoir of 1913
298

22 The Calculus of Infinitesimal Transformations
51
23 Function Groups
56
24 The Invariant Theory of Contact Transformations
62
25 The Birth of Lies Theory of Groups
68
Lies Theory of Transformation Groups 18741893
75
32 An Overview of Lies Theory
79
33 The Adjoint Group
87
34 Complete Systems and Lies Idee Fixe
92
35 The Symplectic Groups
96
The Background to Killings Work on Lie Algebras
100
41 NonEuclidean Geometry and Weierstrassian Mathematics
101
18671872
103
43 NonEuclidean Geometry and General Space Forms
111
44 From Space Forms to Lie Algebras
118
45 Riemann and Helmholtz
124
46 Killing and Klein on the Scope of Geometry
130
Killing and the Structure of Lie Algebras
138
51 Space Forms and Characteristic Equations
139
52 Encounter with Lies Theory
146
53 Correspondence with Engel
150
54 Killings Theory of Structure
156
55 Groups of Rank Zero
168
56 The Lobachevsky Prize
179
The Doctoral Thesis of Elie Cartan
182
61 Lie and the Mathematicians of Paris
183
62 Cartans Theory of Semisimple Algebras
196
63 Killings Secondary Roots
210
64 Cartans Application of Secondary Roots
218
Lies School and Linear Representations
225
71 Representations in Lies Research Program
226
72 Eduard Study
235
85 The Memoirs of 1914
304
The Gottingen School of Hilbert
317
91 Hilbert and the Theory of Invariants
318
92 Hilbert at Gottingen
324
93 The Mathematization of Physics at Gottingen
333
Integral Equations
347
Riemann Surfaces
352
96 Hilberts Brand of Mathematical Thinking
366
The Berlin Algebraists Probenius and I Schur
372
Representations
373
102 Hurwitz and the Theory of Invariants
384
103 Schurs Doctoral Dissertation
394
104 Schurs Career 19011923
402
105 Cayleys Counting Problem Revisited
414
Prom Relativity to Representations
420
112 The Space Problem Reconsidered
432
113 Tensor Algebra and Tensor Symmetries
440
114 Weyls Response to Study
448
115 The GroupTheoretic Foundation of Tensor Calculus
455
Weyls Great Papers of 1925 and 1926
465
122 Schur and the Origins of Weyls 1925 Paper
472
123 Weyls Extension of the KillingCartan Theory
477
124 Weyls Finite Basis Theorem
485
125 Weyls Theory of Characters
487
126 Cartans Response
493
127 The PeterWeyl Paper
500
Suggested Further Reading
513
Published and Unpublished Sources
515
Index
547
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