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

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.