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critically examining Cartan's doctoral attempts to rigorously classify simple Lie algebras, including the use of many unpublished letters.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.
- Discusses the origins of the theory of moving frames and the geometrical theory of Lie groups
- Reviews Cartan's revolutionary contributions to Lie group theory and differential geometry
- Evaluates many unpublished sources that shed light on important aspects of the historical development of Lie algebras
1. Lie on the backstage 2. Cartan's doctoral dissertation 3. Infinite Continuous Groups 1883-1902 4. Exterior Differential Systems 5. Cartan's Theory (1902-1909) 6. The method of moving frames 7. The geometry of continuous groups 8. Conclusion
Alberto Cogliati, currently Assistant Professor at the Department of Mathematics 'F. Enriques' of the Università degli Studi di Milano. He received his PhD in March 2012 discussing Cartan's structural approach to the theory of continuous groups of transformations. His main research interest is the history of 19th-20th mathematics, especially differential geometry, the history of Lie groups and the theory of connections.