| Description |
1 online resource |
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text txt rdacontent |
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computer c rdamedia |
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online resource cr rdacarrier |
| Series |
Studies in the history of mathematical enquiry |
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Studies in the history of mathematical enquiry.
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| Bibliography |
Includes bibliographical references and index. |
| Note |
Online resource; title from PDF title page (EBSCO, viewed October 27, 2017). |
| Summary |
Writing Small Omegas: Elie Cartan's Contributions to the Theory of Continuous Groups 1894-1926 provides a general account of Lie's theory of finite continuous groups, critically examining Cartan's doctoral attempts to rigorously classify simple Lie algebras, including the use of many unpublished letters. It evaluates pioneering attempts to generalize Lie's classical ideas to the infinite-dimensional case in the works of Lie, Engel, Medolaghi and Vessiot. Within this context, Cartan's groundbreaking contributions in continuous group theory, particularly in his characteristic and unique recourse to exterior differential calculus, are introduced and discussed at length. The work concludes by discussing Cartan's contributions to the structural theory of infinite continuous groups, his method of moving frames, and the genesis of his geometrical theory of Lie groups. |
| Contents |
Machine generated contents note: 1. Lie on the Backstage -- 1.1. Fundamentals of Lie Theory of Finite Continuous Groups -- 1.1.1. Three Fundamental Theorems -- 1.1.2. The Adjoint Croup -- 1.2. "A Fundamental Discipline" -- 2. Cartan's Doctoral Dissertation -- 2.1. Finite Continuous Croups -- 2.1.1. Reduced Form of a Given Group -- 2.1.2. Solvability and Semisimplicity Criteria -- 2.1.3. Radical and Decomposition Theorems -- 2.2. Lie Theory of Complete Systems -- 2.3. Complete Systems and Canonical Reduction -- 2.4. Appendix -- 3. Infinite Continuous Groups 1883 -- 1902 -- 3.1. Lie's First Contributions -- 3.2. Differential Invariants -- 3.3. Engel's Habilitationsschrift -- 3.4. Foundations of Infinite Continuous Groups -- 3.5. On a Theorem by Engel -- 3.6. Medolaghi's Contributions -- 3.7. Vessiot and His Memoire couronnee -- 4. Exterior Differential Systems -- 4.1. Some Technical Preliminaries -- 4.2. The State-of-the-Art in the Early 1890s -- 4.3. Engel's Invariants Theory of Pfaffian Systems -- 4.3.1. Invariant Correspondences -- 4.4. Von Weber's Contributions: 1898 -- 1900 -- 4.4.1. Character and Characteristic Transformations -- 4.4.2. Pfaffian Systems of Character One -- 4.4.3. Reducibility of a Pfaffian System to Its Normal Form -- 4.5. The Foundations of the Exterior Differential Calculus -- 4.6. Cartan's Theory of General Pfaffian Systems -- 4.6.1. Geometrical Representation -- 4.6.2. Cauchy's First Theorem -- 4.6.3. Genre and Characters -- 4.6.4. Characteristic Elements -- 4.6.5. Pfaffian Systems of Character One, II -- 4.7. Some Final Remarks -- 5. Cartan's Theory 1902 -- 1909 -- 5.1. On the Genesis of the Theory -- 5.2. Some Examples -- 5.3. Cartan's Theory -- 5.3.1. First Fundamental Theorem -- 5.3.2. Second and Third Fundamental Theorems -- 5.4. Subgroups of a Given Continuous Group -- 5.5. Simple Infinite Continuous Groups -- 5.6. Essential and Inessential Invariants -- 5.7. Some Final Remarks -- 6. Cartan as a Geometer -- 6.1. Introduction -- 6.2. Maurer -- (Cotton) -- Cartan Forms -- 6.3. Cartan's 1910 Paper -- 6.4. The Generalization of the Notion of Space -- 6.5. Cartan's Collaboration with Schouten -- 6.6. Concluding Remarks -- A. Picard -- Vessiot Theory -- B. The Galois of His Generation -- C. Clifford's Parallelism -- C.1. Klein's Zur Nicht-Euklidischen Geometrie -- C.2. Bianchi and Fubini -- C.3. Enea Bortolotti. |
| Subject |
Continuous groups.
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Groupes continus.
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MATHEMATICS -- Algebra -- Intermediate.
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Continuous groups
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| Other Form: |
Print version: Cogliati, Alberto. Writing small omegas : elie cartan's contributions to the theory of continuous groups 1894-1926. First edition. London, England : Academic Press, ©2018 approximately 99 pages 9780128142448 |
| ISBN |
9780128142745 (electronic bk.) |
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012814274X (electronic bk.) |
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0128142448 |
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9780128142448 |
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9780128142448 |
| Standard No. |
AU@ 000061502896 |
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AU@ 000065067038 |
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AU@ 000068211123 |
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