Wilhelm Killing
Wilhelm Karl Joseph Killing



Born  10 May 1847 
Died  11 February 1923 (19230212) (aged 75) 
Residence  Germany 
Citizenship  German 
Known for  Lie algebras, Lie groups, and nonEuclidean geometry 
Awards  Lobachevsky Prize (1900) 
Scientific career  
Fields  Mathematics 
Doctoral advisor  Karl Weierstrass Ernst Kummer 
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and nonEuclidean geometry.
Life
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 Braunsberg (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.
Work
In 1878 Killing wrote on space forms in terms of nonEuclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885.^{[1]} Recounting lectures of Weierstrass, he there introduced the hyperboloid model of hyperbolic geometry described by Weierstrass coordinates.^{[2]} He is also credited with formulating transformations mathematically equivalent to Lorentz transformations in n dimensions in 1885,^{[3]} see History of Lorentz transformations#Killing.
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.^{[citation needed]}
From 1888 to 1890, Killing essentially classified the complex finitedimensional simple Lie algebras, as a requisite step of classifying Lie groups, inventing the notions of a Cartan subalgebra and the Cartan matrix. He thus arrived at the conclusion that, basically, the only simple Lie algebras were those associated to the linear, orthogonal, and symplectic groups, apart from a small number of isolated exceptions. Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper. Killing also introduced the notion of a root system. He discovered the exceptional Lie algebra g_{2} 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."^{[4]}
See also
 Killing equation
 Killing form
 Killing–Hopf theorem
 Killing horizon
 Killing spinor
 Killing tensor
 Killing vector field
References
 Work on nonEuclidean geometry
 Killing, W. (1878) [1877]. "Ueber zwei Raumformen mit constanter positiver Krümmung". Journal für die reine und angewandte Mathematik. 86: 72–83.
 Killing, W. (1880) [1879]. "Die Rechnung in den NichtEuklidischen Raumformen". Journal für die reine und angewandte Mathematik. 89: 265–287.
 Killing, W. (1885) [1884]. "Die Mechanik in den NichtEuklidischen Raumformen". Journal für die reine und angewandte Mathematik. 98: 1–48.
 Killing, W. (1885). Die nichteuklidischen Raumformen. Leipzig: Teubner.
 Killing, W. (1891). "Ueber die CliffordKlein'schen Raumformen". Mathematische Annalen. 39: 257–278.
 Killing, W. (1892). "Ueber die Grundlagen der Geometrie". Journal für die reine und angewandte Mathematik. 109: 121–186.
 Killing, W. (1893). "Zur projectiven Geometrie". Mathematische Annalen. 43: 569–590.
 Killing, W. (1893). Einführung in die Grundlagen der Geometrie I. Paderborn: Schöningh.
 Killing, W. (1898) [1897]. Einführung in die Grundlagen der Geometrie II. Paderborn: Schöningh.
 Work on transformation groups
 Killing, W. (1888). "Die Zusammensetzung der stetigen endlichen Transformationsgruppen". Mathematische Annalen. 31: 252–290.
 Killing, W. (1889). "Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Zweiter Theil". Mathematische Annalen. 33: 1–48.
 Killing, W. (1889). "Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Dritter Theil". Mathematische Annalen. 34: 57–122.
 Killing, W. (1890). "Erweiterung des Begriffes der Invarianten von Transformationsgruppen". Mathematische Annalen. 35: 423–432.
 Killing, W. (1890). "Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Vierter Theil". Mathematische Annalen. 36: 161–189.
 Killing, W. (1890). "Bestimmung der grössten Untergruppen von endlichen Transformationsgruppen". Mathematische Annalen. 36: 239–254.
External links
 O'Connor, John J.; Robertson, Edmund F., "Wilhelm Killing", MacTutor History of Mathematics archive, University of St Andrews .
Media related to Wilhelm Killing (mathematician) at Wikimedia Commons
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