Lie groups and Lie algebras arose in the study of the mathematical properties of rotations. The ideas were originally developed by the Norwegian mathematician Sophus Lie (1849-1899), and they play an important role in the development of 20th century mathematics and mathematical physics. This book focuses on the origins and subsequent development of the structure of the theory, particularly the structure and representations of semisimple groups, with particular emphasis on the motivation behind the mathematics. It should thus be useful not only to historians of mathematics and to mathematicians interested in the history of their field, but also to students.

Klappentext

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.

Inhalt

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.