Emergence of the theory of Lie groups : an essay in the history of mathematics, 1869-1926
著者
書誌事項
Emergence of the theory of Lie groups : an essay in the history of mathematics, 1869-1926
（Sources and studies in the history of mathematics and physical sciences）
Springer, c2000
- : hbk.
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
The great Norwegian mathematician Sophus Lie developed the general theory of transformations in the 1870s, and the first part of the book properly focuses on his work. In the second part the central figure is Wilhelm Killing, who developed structure and classification of semisimple Lie algebras. The third part focuses on the developments of the representation of Lie algebras, in particular the work of Elie Cartan. The book concludes with the work of Hermann Weyl and his contemporaries on the structure and representation of Lie groups which serves to bring together much of the earlier work into a coherent theory while at the same time opening up significant avenues for further work.
目次
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 Idee 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 Idee 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: Elie Cartan.- 6. The Doctoral Thesis of Elie 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 Goettingen School of Hilbert.- 9.1. Hilbert and the Theory of Invariants.- 9.2. Hilbert at Goettingen.- 9.3. The Mathematization of Physics at Goettingen ..- 9.4. Weyl's Goettingen Years: Integral Equations.- 9.5. Weyl's Goettingen 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.
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