Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926 (Sources and Studies in the History of Mathematics and Physical Sciences)

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](https://www.maa.org/programs/maa-awards/writing-awards/chauvenet-prizes) for mathematical exposition **[Hawkins, Thomas. “**[ **The Birth of Lie’s Theory of Groups**](https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/9BK79TZI/Hawkins%20-%201994%20-%20The%20birth%20of%20Lie's%20theory%20of%20groups.pdf) **.”** The Mathematical Intelligencer **16, 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](http://ams.rice.edu/mathscinet/search/author.html?mrauthid=353912) * * * cf. Hawkins's [*Episodes in the Origins of the Representation Theory of Lie Algebras*](https://www.youtube.com/watch?v=qMN_XzTqzao) This book is both more and less than a history of the theory of Lie groups during the period 1869-1926. No attempt has been made to provide an exhaustive treatment of all aspects of the theory. Instead, I have focused upon its origins and upon the subsequent development of its structural as­ pects, particularly the structure and representation of semisimple groups. In dealing with this more limited subject matter, considerable emphasis has been placed upon the motivation behind the mathematics. This has meant paying close attention to the historical context: the mathematical or physical considerations that motivate or inform the work of a particular mathematician as well as the disciplinary ideals of a mathematical school that encourage research in certain directions. As a result, readers will ob­ tain in the ensuing pages glimpses of and, I hope, the flavor of many areas of nineteenth and early twentieth century geometry, algebra, and analysis. They will also encounter many of the mathematicians of the period, includ­ ing quite a few not directly connected with Lie groups, and will become acquainted with some of the major mathematical schools. In this sense, the book is more than a history of the theory of Lie groups. It provides a different perspective on the history of mathematics between, roughly, 1869 and 1926. Hence the subtitle. The Great Norwegian Mathematician Sophus Lie Developed The General Theory Of Transformations In The 1870s, And The First Part Of The Book Properly Focuses On His Work. In The Second Part The Central Figure Is Wilhelm Killing, Who Developed Structure And Classification Of Semisimple Lie Algebras. The Third Part Focuses On The Developments Of The Representation Of Lie Algebras, In Particular The Work Of Elie Cartan. The Book Concludes With The Work Of Hermann Weyl And His Contemporaries On The Structure And Representation Of Lie Groups Which Serves To Bring Together Much Of The Earlier Work Into A Coherent Theory While At The Same Time Opening Up Significant Avenues For Further Work. Preface The Geometrical Origins of Lie's theory Jacobi & The Analytical Origins of Lie's Theory Lie's Theory of Transformation Groups 1874-1893 Non-euclidean Geometry & Weierstrassian Mathematics Killing & the Structure of Lie Algebras The Doctoral Thesis of Elie Cartan Lie's School & Linear Representations Cartan's Trilogy: 1913-14 The Göttingen School of Hilbert The Berlin Algebraists: Frobenius & Schur From Relativity to Representations Weyl's Great Papers of 1925 & 1926 References Index.