Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869-1926
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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The Geometrical Origins of Lies Theory
11 Tetrahedral Line Complexes
12 WCurves and WSurfaces
13 Lies Idee Fixe
14 The Sphere Mapping
15 The Erlanger Programm
Jacobi and the Analytical Origins of Lies Theory
21 Jacobis Two Methods
73 Gino Fano
74 Cayleys Counting Problem
75 Kowalewskis Theory of Weights
Cartans Trilogy 191314
81 Research Priorities 18931909
82 Another Application of Secondary Roots
83 Continuous Groups and Geometry
84 The Memoir of 1913
22 The Calculus of Infinitesimal Transformations
23 Function Groups
24 The Invariant Theory of Contact Transformations
25 The Birth of Lies Theory of Groups
Lies Theory of Transformation Groups 18741893
32 An Overview of Lies Theory
33 The Adjoint Group
34 Complete Systems and Lies Idee Fixe
35 The Symplectic Groups
The Background to Killings Work on Lie Algebras
41 NonEuclidean Geometry and Weierstrassian Mathematics
43 NonEuclidean Geometry and General Space Forms
44 From Space Forms to Lie Algebras
45 Riemann and Helmholtz
46 Killing and Klein on the Scope of Geometry
Killing and the Structure of Lie Algebras
51 Space Forms and Characteristic Equations
52 Encounter with Lies Theory
53 Correspondence with Engel
54 Killings Theory of Structure
55 Groups of Rank Zero
56 The Lobachevsky Prize
The Doctoral Thesis of Elie Cartan
61 Lie and the Mathematicians of Paris
62 Cartans Theory of Semisimple Algebras
63 Killings Secondary Roots
64 Cartans Application of Secondary Roots
Lies School and Linear Representations
71 Representations in Lies Research Program
72 Eduard Study
85 The Memoirs of 1914
The Gottingen School of Hilbert
91 Hilbert and the Theory of Invariants
92 Hilbert at Gottingen
93 The Mathematization of Physics at Gottingen
96 Hilberts Brand of Mathematical Thinking
The Berlin Algebraists Probenius and I Schur
102 Hurwitz and the Theory of Invariants
103 Schurs Doctoral Dissertation
104 Schurs Career 19011923
105 Cayleys Counting Problem Revisited
Prom Relativity to Representations
112 The Space Problem Reconsidered
113 Tensor Algebra and Tensor Symmetries
114 Weyls Response to Study
115 The GroupTheoretic Foundation of Tensor Calculus
Weyls Great Papers of 1925 and 1926
122 Schur and the Origins of Weyls 1925 Paper
123 Weyls Extension of the KillingCartan Theory
124 Weyls Finite Basis Theorem
125 Weyls Theory of Characters
126 Cartans Response
127 The PeterWeyl Paper
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