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

515 | |

547 | |

### 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