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

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Springer Science & Business Media, Aug 1, 2000 - Mathematics - 564 pages
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Written by the recipient of the 1997 MAA Chauvenet Prize for mathematical exposition, this book tells how the theory of Lie groups emerged from a fascinating cross fertilization of many strains of 19th and early 20th century geometry, analysis, mathematical physics, algebra and topology. The reader will meet a host of mathematicians from the period and become acquainted with the major mathematical schools. The first part describes the geometrical and analytical considerations that initiated the theory at the hands of the Norwegian mathematician, Sophus Lie. The main figure in the second part is Weierstrass'student Wilhelm Killing, whose interest in the foundations of non-Euclidean geometry led to his discovery of almost all the central concepts and theorems on the structure and classification of semisimple Lie algebras. The scene then shifts to the Paris mathematical community and Elie Cartans work on the representation of Lie algebras. The final part describes the influential, unifying contributions of Hermann Weyl and their context: Hilberts Göttingen, general relativity and the Frobenius-Schur theory of characters. The book is written with the conviction that mathematical understanding is deepened by familiarity with underlying motivations and the less formal, more intuitive manner of original conception. The human side of the story is evoked through extensive use of correspondence between mathematicians. The book should prove enlightening to a broad range of readers, including prospective students of Lie theory, mathematicians, physicists and historians and philosophers of science.
  

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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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About the author (2000)

Thomas Hawkins is co-pastor with his wife, Jan, at First Presbyterian Church in Charleston, Illinois. Prior to this pastorate, Hawkins was professor in the Career and Organizational Studies Program, Lumpkin College of Business and Applied Sciences at Eastern Illinois University. He is the author of several books published by Discipleship Resources, including

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