Emergence of the Theory of Lie Groups

by Hawkins, Thomas; Buchwald, Jed Z.; Lutzen, J.; Toomer, G. J.

• ISBN13: 9780387989631
• ISBN10: 0387989633
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Springer Verlag

Summary

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 Cartan's work on the representation of Lie algebras. The final part describes the influential, unifying contributions of Hermann Weyl and their context: Hilbert's 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.

Table of Contents

Preface	v
Part I: Sophus Lie
The Geometrical Origins of Lie's Theory	1	(42)
Tetrahedral Line Complexes	2	(8)
W-Curves and W-Surfaces	10	(10)
Lie's Idee Fixe	20	(6)
The Sphere Mapping	26	(8)
The Erlanger Programm	34	(9)
Jacobi and the Analytical Origins of Lie's Theory	43	(32)
Jacobi's Two Methods	44	(7)
The Calculus of Infinitesimal Transformations	51	(5)
Function Groups	56	(6)
The Invariant Theory of Contact Transformations	62	(6)
The Birth of Lie's Theory of Groups	68	(7)
Lie's Theory of Transformation Groups 1874--1893	75	(25)
The Group Classification Problem	75	(4)
An Overview of Lie's Theory	79	(8)
The Adjoint Group	87	(5)
Complete Systems and Lie's Idee Fixe	92	(4)
The Symplectic Groups	96	(4)
Part II: Wilhelm Killing
The Background to Killing's Work on Lie Algebras	100	(38)
Non-Euclidean Geometry and Weierstrassian Mathematics	101	(2)
Student Years in Berlin: 1867--1872	103	(8)
Non-Euclidean Geometry and General Space Forms	111	(7)
From Space Forms to Lie Algebras	118	(6)
Riemann and Helmholz	124	(6)
Killing and Klein on the Scope of Geometry	130	(8)
Killing and the Structure of Lie Algebrass	138	(44)
Spaces Forms and Characteristic Equations	139	(7)
Encounter with Lie's Theory	146	(4)
Correspondence with Engel	150	(6)
Killing's Theory of Structure	156	(12)
Groups of Rank Zero	168	(11)
The Lobachevsky Prize	179	(3)
Part III: Elie Cartan
The Doctoral Thesis of Elie Cartan	182	(43)
Lie and the Mathematicians of Paris	183	(13)
Cartan's Theory of Semisimple Algebras	196	(14)
Killing's Secondary Roots	210	(8)
Cartan's Application of Secondary Roots	218	(7)
Lie's School & Linear Representations	225	(52)
Representations in Lie's Research Program	226	(9)
Eduard Study	235	(16)
Gino Fano	251	(9)
Cayley's Counting Problem	260	(5)
Kowalewski's Theory of Weights	265	(12)
Cartan's Trilogy: 1913--14	277	(40)
Research Priorities 1893--1909	278	(9)
Another Application of Secondary Roots	287	(3)
Continuous Groups and Geometry	290	(8)
The Memoir of 1913	298	(6)
The Memoirs of 1914	304	(13)
Part IV: Hermann Weyl
The Gottingen School of Hilbert	317	(55)
Hilbert and the Theory of Invariants	318	(6)
Hilbert at Gottingen	324	(9)
The Mathematization of Physics at Gottingen	333	(14)
Weyl's Gottingen Years: Integral Equations	347	(5)
Weyl's Gottingen Years: Riemann Surfaces	352	(14)
Hilbert's Brand of Mathematical Thinking	366	(6)
The Berlin Algebraists: Frobenius & Schur	372	(48)
Frobenius' Theory of Group Characters & Representations	373	(11)
Hurwitz and the Theory of Invariants	384	(10)
Schur's Doctoral Dissertation	394	(8)
Schur's Career 1901--1923	402	(12)
Cayley's Counting Problem Revisited	414	(6)
From Relativity to Representations	420	(45)
Einstein's General Theory of Relativity	420	(12)
The Space Problem Reconsidered	432	(8)
Tensor Algebra & Tensor Symmetries	440	(8)
Weyl's Response to Study	448	(7)
The Group-Theoretic Foundation of Tensor Calculus	455	(10)
Weyl's Great Papers of 1925 and 1926	465	(48)
The Complete Reducibility Theorem	465	(7)
Schur and the Origins of Weyl's 1925 Paper	472	(5)
Weyl's Extension of the Killing-Cartan Theory	477	(8)
Weyl's Finite Basis Theorem	485	(2)
Weyl's Theory of Characters	487	(6)
Cartan's Response	493	(7)
The Peter-Weyl Paper	500	(13)
Afterword. Suggested Further Reading	513	(2)
References. Published & Unpublished Sources	515	(32)
Index	547