## Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926This 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. |

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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 |

515 | |

547 | |

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Emergence of the Theory of Lie Groups: An Essay in the History of ... Thomas Hawkins Limited preview - 2012 |

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### Common terms and phrases

adjoint group application Berlin Cartan Chapter classification coefficients complete reducibility theorem consider contact transformations continuous groups coordinates corresponding covariant curves Darboux defined denote developed dissertation elements Engel Erlan9er Pro9ramm finite groups follows Frobenius Frobenius's theory function fundamental Galois Gottingen Hilbert homogeneous homogeneous coordinates Hurwitz ideas infinitesimal transformations irreducible representations isomorphic Jacobi Killing Killing's Klein Kowalewski leave nothing planar lectures Leipzig letter Lie algebra Lie groups Lie's theory line geometry linear transformations manifold Math mathematical mathematicians method non-Euclidean geometry notation orthogonal paper Paris partial differential equations Picard planar invariant Poincare polynomial problem of determining projective groups proof quadratic rank zero realized Reprinted in Abhandlun9en Riemann satisfying Schur secondary roots Section semisimple groups simple groups solution space forms structure Study's subgroup surface symmetric tensor theory of groups theory of invariants tion transformation groups unitarian trick variables vector Weierstrass weight Weyl Weyl's