Wilhelm Killing

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Wilhelm Karl Joseph Killing
Born	May 10, 1847
Died	February 11, 1923
Residence	Germany
Citizenship	German
Fields	Mathematics
Known for	Lie algebras, Lie groups, and non-Euclidean geometry

Wilhelm Karl Joseph Killing (May 10, 1847– February 11, 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.

Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. He became a professor at the seminary college Collegium Hosianum in Braumsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town council. As a professor and administrator Killing was widely liked and respected. Finally, in 1892 he became professor at the University of Münster. Killing and his spouse had entered the Third Order of Franciscans in 1886.

Killing invented Lie algebras independently of Sophus Lie around 1880. Killing's university library did not contain the Scandinavian journal in which Lie's article appeared. (Lie later was scornful of Killing, perhaps out of competitive spirit and claimed that all that was valid had already been proven by Lie and all that was invalid was added by Killing.) In fact Killing's work was less rigorous logically than Lie's, but Killing had much grander goals in terms of classification of groups, and made a number of unproven conjectures that turned out to be true. Because Killing's goals were so high, he was excessively modest about his own achievement.

Killing (1888-1890) essentially classified the complex simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix. Elie Cartan's dissertation was essentially a rigorous re-writing of Killing's paper. Killing also introduced the notion of a root system. He is the discoverer of the exceptional Lie algebra g₂ (in 1887); his root system classification showed up all the exceptional cases, but concrete constructions came later.

As A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."

Killing also introduced the term characteristic equation of a matrix.

[edit] References

Coleman, A. John, "The Greatest Mathematical Paper of All Time," The Mathematical Intelligencer, vol. 11, no. 3, pp. 29-38.
Hawkins, Thomas, Emergence of the Theory of Lie Groups, New York: Springer, 2000.
Killing, Die Zusammensetzung der stetigen/endlichen Transformationsgruppen Mathematische Annalen, Volume 31, Number 2 June, 1888, Pages 252-290 doi:10.1007/BF01211904, Volume 33, Number 1 March, 1888, Pages 1-48 doi:10.1007/BF01444109, Volume 34, Number 1 March, 1889, Pages 57-122 doi:10.1007/BF01446792, Volume 36, Number 2 June, 1890,Pages 161-189 doi:10.1007/BF01207837