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

Description

Klein and Lie "were self-styled 'synthesists' in the midst of analysts and arithmeticians" (p. 3 // PDF p. 16)

This book doesn't even cite Lie's 1891 work on differential equations!

Written by the recipient of the 1997 MAA Chauvenet Prize for mathematical exposition [Hawkins, Thomas. “The Birth of Lie’s Theory of Groups.” The Mathematical Intelligencer16, no. 2 (March 1, 1994): 6–17.] , this book tells how the theory of Lie groups emerged from a fascinating cross fertilization of many strains of 19th and early 20th century geometry, analysis, mathematical physics, algebra and topology. The reader will meet a host of mathematicians from the period and become acquainted with the major mathematical schools. The first part describes the geometrical and analytical considerations that initiated the theory at the hands of the Norwegian mathematician, Sophus Lie. The main figure in the second part is Weierstrass'student Wilhelm Killing, whose interest in the foundations of non-Euclidean geometry led to his discovery of almost all the central concepts and theorems on the structure and classification of semisimple Lie algebras. The scene then shifts to the Paris mathematical community and Elie Cartans work on the representation of Lie algebras. The final part describes the influential, unifying contributions of Hermann Weyl and their context: Hilberts Göttingen, general relativity and the Frobenius-Schur theory of characters. The book is written with the conviction that mathematical understanding is deepened by familiarity with underlying motivations and the less formal, more intuitive manner of original conception. The human side of the story is evoked through extensive use of correspondence between mathematicians. The book should prove enlightening to a broad range of readers, including prospective students of Lie theory, mathematicians, physicists and historians and philosophers of science.

The book under review is a very nice essay on the history of the theory of Lie groups during the period 1869–1926. It is focused upon the origins of the theory and on the subsequent developments of its structural aspects, particularly the structure and representations of semisimple groups.
The book is divided into four parts, each bearing the name of a mathematician, who stands out as the central figure there. The first part is devoted to the geometrical and analytical origins of the theory of continuous transformation groups of Sophus Lie—the precursor of the modern theory of Lie groups. In the second part the central figure is Wilhelm Killing, who discovered almost all central concepts and theorems on the structure and classification of semisimple Lie algebras. The third part is named after Élie Cartan and is primarily concerned with developments that would now be interpreted as representations of Lie algebras, particularly simple and semisimple algebras. In the last part the main role is played by Hermann Weyl and this part itself is mainly focused on the development of representation theory of Lie groups and algebras.
The book has 50 illustrations and includes quite a long list of historical references and a substantial index. Reviewed by Volodymyr Mazorchuk

cf. Hawkins's Episodes in the Origins of the Representation Theory of Lie Algebras