The greatest mathematical paper of all time
type: Item
instance of: scholarly article
title: The greatest mathematical paper of all time
zbMATH Open document ID: 0683.01007
DOI: 10.1007/BF03025189
published in: The Mathematical Intelligencer
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 01A70
Mathematics Subject Classification ID: 22-03
zbMATH DE Number: 4118317
zbMATH Keywords: Wilhelm Killing
zbMATH Keywords: Elie Cartan
zbMATH Keywords: linear associative algebra
zbMATH Keywords: Lie algebras
zbMATH Keywords: I. Macdonald
zbMATH Keywords: V. Kac
zbMATH Keywords: R. Moody
zbMATH Keywords: Weyl groups
zbMATH Keywords: Coxeter transformation
zbMATH Keywords: Super-String theory
Wikidata QID: Q56211628
author: A. John Coleman
reviewed by: Cristina Irimia
MaRDI profile type: Publication
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cites work: On Infinite Root Systems
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full work available at URL: https://doi.org/10.1007/bf03025189
OpenAlex ID: W2133875637
description: scientific article; zbMATH DE number 4118317