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**Emergence of the theory of Lie groups. An essay in the history of mathematics 1869–1926.**
*(English)*
Zbl 0965.01001

Sources and Studies in the History of Mathematics and Physical Sciences. New York, NY: Springer. xiii, 564 p. (2000).

The subtitle gives the context in which this book should be read. The author’s purpose was to use the development of the theory of Lie groups as a vehicle to unify a number of separate essays on the history of mathematics in the late nineteenth and early twentieth centuries. Each of its four parts, which can be read independently of one another, is in turn unified by the works of a particular mathematician. The four mathematicians singled out for this honor are Sophus Lie, Wilhelm Killing, Élie Cartan, and Hermann Weyl. Although these four play the leading role in their parts, they by no means monopolize the exposition; and one finds in each part a fairly detailed exposition of the key moments in the era under consideration.

Part I begins with the prehistory of the subject of Lie groups, its geometric and analytic sources, with special emphasis on the work of the nineteenth-century geometers, apparently communicated to Lie – or at least advocated – by Felix Klein, and the work of Jacobi in partial differential equations. This part ends with an exposition of the theory in the original form given it by Lie himself, which is really confined to what we would now call a neighborhood of the identity, and, as the author notes, really defines a semi-group rather than a group.

Klein also figures importantly in Part 2, because of the contrast between his “generic” approach to geometry and Killing’s algebraic approach, with which Klein had little sympathy, regarding it as replete with degenerate cases. The greater generality that Killing insisted upon took the theory of Lie algebras in a new direction not intended by Lie. The two were in correspondence, but Killing seems to have paid much more attention to Lie’s work than Lie did to his, and it was Lie’s disciple Engel with whom Killing had the more fruitful exchange of ideas.

Killing left a number of interesting conjectures, which the author discusses in Part 3. Élie Cartan became interested in continuous groups while still a student, and spent considerable time assimilating, justifying, and extending the work of Killing. (He met Lie, but, according to the author, was very little influenced by Lie’s methods. Where Lie differs from Killing, Cartan’s work takes off from Killing’s rather than Lie’s.) Part 3 recounts a large variety of topics in the development of representation theory from the 1880s through the early twentieth century, culminating in Cartan’s 1914 classification of the real projective groups of real projective space that leave nothing planar invariant.

Part 4, which is the longest part, belongs mostly to the Göttingen mathematicians of Hilbert’s school, and, of course, the main character is Hermann Weyl. This section contains a great deal of material also on the mathematization of physics, especially relativity theory. It carries the story down to Weyl’s papers of 1925-26, which contained his theory of characters, and Cartan’s response to them.

The book, which represents the well-organized results of many years of labor on the part of the author, is clearly written, profusely illustrated, thoroughly indexed, and provided with an extensive set of references.

Part I begins with the prehistory of the subject of Lie groups, its geometric and analytic sources, with special emphasis on the work of the nineteenth-century geometers, apparently communicated to Lie – or at least advocated – by Felix Klein, and the work of Jacobi in partial differential equations. This part ends with an exposition of the theory in the original form given it by Lie himself, which is really confined to what we would now call a neighborhood of the identity, and, as the author notes, really defines a semi-group rather than a group.

Klein also figures importantly in Part 2, because of the contrast between his “generic” approach to geometry and Killing’s algebraic approach, with which Klein had little sympathy, regarding it as replete with degenerate cases. The greater generality that Killing insisted upon took the theory of Lie algebras in a new direction not intended by Lie. The two were in correspondence, but Killing seems to have paid much more attention to Lie’s work than Lie did to his, and it was Lie’s disciple Engel with whom Killing had the more fruitful exchange of ideas.

Killing left a number of interesting conjectures, which the author discusses in Part 3. Élie Cartan became interested in continuous groups while still a student, and spent considerable time assimilating, justifying, and extending the work of Killing. (He met Lie, but, according to the author, was very little influenced by Lie’s methods. Where Lie differs from Killing, Cartan’s work takes off from Killing’s rather than Lie’s.) Part 3 recounts a large variety of topics in the development of representation theory from the 1880s through the early twentieth century, culminating in Cartan’s 1914 classification of the real projective groups of real projective space that leave nothing planar invariant.

Part 4, which is the longest part, belongs mostly to the Göttingen mathematicians of Hilbert’s school, and, of course, the main character is Hermann Weyl. This section contains a great deal of material also on the mathematization of physics, especially relativity theory. It carries the story down to Weyl’s papers of 1925-26, which contained his theory of characters, and Cartan’s response to them.

The book, which represents the well-organized results of many years of labor on the part of the author, is clearly written, profusely illustrated, thoroughly indexed, and provided with an extensive set of references.

Reviewer: Roger Cooke (Burlington)