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Essays in the history of Lie groups and algebraic groups. (English) Zbl 1087.01011

History of Mathematics (Providence) 21. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0288-7/hbk). xiii, 184 p. (2001).
Lie groups and Lie algebras are widely considered to be among the most satisfactory and aesthetic theories in mathematics. The book under review is of high scientific level; it describes the historical evolution during the ‘first century’ of the subject following S. Lie’s work in 1873. It is written by an outstanding contributor to the development of the theory. It presents six chapters corresponding to major publications each of them provided with explanatory remarks and complements, as well as historical notes.
A preliminary chapter features the links between the various topics and the general framework; another introductory chapter is about the full reducibility of the Lie group \(\text{SL}(2,\mathbb C).\)
The first chapter presents H. Weyl’s achievements, their impact on É. Cartan and their contributions to the uprise of harmonic analysis and quantum mechanics. The next chapter concerns É. Cartan’s study of general semisimple Lie groups and his theory of symmetric spaces, i.e., homogeneous Riemannian manifolds.
The second part of the book is devoted to globalization processes. First comes an extensive presentation of linear algebraic groups during the 19th century, with references to S. Lie, E. Study, É. Picard, É. Cartan, K. Carda, and especially L. Maurer. During the 20th century the subject is revived by C. Chevalley and E. Kolchin in the 1940s and then by J. Tits. A supplementary specific chapter presents the work of C. Chevalley in Lie groups and linear algebraic groups. The last chapter is about the work of E. Kolchin and his development of a Galois theory for strongly normal extensions, relating the subject to the initial results obtained by Galois during his studies of differential equations.
The book is an essential contribution to the comprehension of the central position taken by Lie group theory. It complements T. Hawkins, “The emergence of the theory of Lie groups. An essay on the history of mathematics, 1869–1926. Springer, Berlin (2000; Zbl 0965.01001). It should be read by any person wanting to learn about the cultural interest of mathematics because it is written by one of the most important actors in the field, who does not hesitate to speak of a whole area that “opens up new directions and points to work for the future”.

MSC:

01A55 History of mathematics in the 19th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A60 History of mathematics in the 20th century
17-03 History of nonassociative rings and algebras
20-03 History of group theory
22-03 History of topological groups

Citations:

Zbl 0965.01001
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