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

ISBN-13: 9780387989631 / Angielski / Twarda / 2000 / 566 str.

Thomas Hawkins

Opis

This 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."

Recenzje

"....this study is just as clearly a stunning achievement. Few historians of mathematics have made a serious attempt to cross the bridge joining the nineteenth and twentieth centuries, and those who have made the journey have tended to avert their eyes from the mainstream traffic....the single greatest merit of Hawkins' book is that the author tries to place the reader in the middle of the action, offering a close up look at how mathematics gets made...Hawkins' account of this strange but wonderful saga resurrects a heroic chapter in the history of mathematics. For anyone with a serious interest in the rich background developments that led to modern Lie theory, this book should be browsed, read, savored, and read again."

-Notices of the AMS

Spis treści

I: Sophus Lie.- 1. The Geometrical Origins of Lie’s Theory.- 1.1. Tetrahedral Line Complexes.- 1.2. W-Curves and W-Surfaces.- 1.3. Lie’s Idée Fixe.- 1.4. The Sphere Mapping.- 1.5. The Erlanger Programm.- 2. Jacobi and the Analytical Origins of Lie’s Theory.- 2.1. Jacobi’s Two Methods.- 2.2. The Calculus of Infinitesimal Transformations.- 2.3. Function Groups.- 2.4. The Invariant Theory of Contact Transformations.- 2.5. The Birth of Lie’s Theory of Groups.- 3. Lie’s Theory of Transformation Groups 1874–1893..- 3.1. The Group Classification Problem.- 3.2. An Overview of Lie’s Theory.- 3.3. The Adjoint Group.- 3.4. Complete Systems and Lie’s Idée Fixe.- 3.5. The Symplectic Groups.- II: Wilhelm Killing.- 4. The Background to Killing’s Work on Lie Algebras.- 4.1. Non-Euclidean Geometry and Weierstrassian Mathematics.- 4.2. Student Years in Berlin: 1867–1872.- 4.3. Non-Euclidean Geometry and General Space Forms.- 4.4. From Space Forms to Lie Algebras.- 4.5. Riemann and Helmholz.- 4.6. Killing and Klein on the Scope of Geometry.- >Chapter 5. Killing and the Structure of Lie Algebrass.- 5.1. Spaces Forms and Characteristic Equations.- 5.2. Encounter with Lie’s Theory.- 5.3. Correspondence with Engel.- 5.4. Killing’s Theory of Structure.- 5.5. Groups of Rank Zero.- 5.6. The Lobachevsky Prize.- III: Élie Cartan.- 6. The Doctoral Thesis of Élie Cartan.- 6.1. Lie and the Mathematicians of Paris.- 6.2. Cartan’s Theory of Semisimple Algebras.- 6.3. Killing’s Secondary Roots.- 6.4. Cartan’s Application of Secondary Roots.- 7. Lie’s School & Linear Representations.- 7.1. Representations in Lie’s Research Program.- 7.2. Eduard Study.- 7.3. Gino Fano.- 7.4. Cayley’s Counting Problem.- 7.5. Kowalewski’s Theory of Weights.- 8. Cartan’s Trilogy: 1913–14.- 8.1. Research Priorities 1893–1909.- 8.2. Another Application of Secondary Roots.- 8.3. Continuous Groups and Geometry.- 8.4. The Memoir of 1913.- 8.5. The Memoirs of 1914.- IV: Hermann Weyl.- 9. The Göttingen School of Hilbert.- 9.1. Hilbert and the Theory of Invariants.- 9.2. Hilbert at Göttingen.- 9.3. The Mathematization of Physics at Göttingen ..- 9.4. Weyl’s Göttingen Years: Integral Equations.- 9.5. Weyl’s Göttingen Years: Riemann Surfaces.- 9.6. Hilbert’s Brand of Mathematical Thinking.- 10. The Berlin Algebraists: Frobenius & Schur.- 10.1. Frobenius’ Theory of Group Characters & Representations.- 10.2. Hurwitz and the Theory of Invariants.- 10.3. Schur’s Doctoral Dissertation.- 10.4. Schur’s Career 1901–1923.- 10.5. Cayley’s Counting Problem Revisited.- 11. From Relativity to Representations.- 11.1. Einstein’s General Theory of Relativity.- 11.2. The Space Problem Reconsidered.- 11.3. Tensor Algebra & Tensor Symmetries.- 11.4. Weyl’s Response to Study.- 11.5. The Group-Theoretic Foundation of Tensor Calculus.- 12. Weyl’s Great Papers of 1925 and 1926.- 12.1. The Complete Reducibility Theorem.- 12.2. Schur and the Origins of Weyl’s 1925 Paper.- 12.3. Weyl’s Extension of the Killing-Cartan Theory.- 12.4. Weyl’s Finite Basis Theorem.- 12.5. Weyl’s Theory of Characters.- 12.6. Cartan’s Response.- 12.7. The Peter-Weyl Paper.- Afterword. Suggested Further Reading.- References. Published & Unpublished Sources.

Blurb

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.