The greatest mathematical paper of all time
The author chooses W. Killing's ``Die Zusammensetzung der stetigen, endlichen Transformationsgruppen'' (Z.v.G. II), dated ``Braunsberg, 2 Februar, 1888'', as the most significant mathematical paper issued in the past 200 years [Math. Ann. 33, 1--48 (1889; JFM 20.0368.03)]. Although the name attached to a key result is that of the follow-up person who exploited an idea or theorem rather that its originator, and most mathematicians seem to have little or no interest in history, no one has suffered from this ahistoricism more than W. Killing. What is now called Lie algebras were invented by the Norwegian Sophus Lie, about 1870 and, independently, by W. Killing about 1880. To reconsider the truth, the author concludes: 1. Killing's paper was the immediate inspiration of the work of Cartan, Molien and Maschke, on the structure of linear associative algebras; 2. Weyl's theory of representation of semisimple Lie groups would have been impossible without ideas, results and methods originated by Killing in Z.v.G. II (loc. cit.); 3. The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody and others, started with Killing; 4. The Weyl groups and the Coxeter transformation are in Z.v.G. II, where they are realized as permutations of the roots; also, the conditions for symmetrisability are given in Z.v.G. II; 5. Killing discovered the Lie algebra \(E_8\), which can be the main hope for saving Super-String theory. 6. About one third of the extraordinary work of Elie Cartan was based, more or less directly, on Z.v.G. II.
scientific article; zbMATH DE number 4118317
scientific article