Wilhelm Killing

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Wilhelm Killing
Wilhelm Killing on the honor roll Lyceum Hosianum in Braniewo

Wilhelm Killing (born May 10, 1847 in Burbach near Siegen , † February 11, 1923 in Münster ) was a German mathematician .


Killing's father was first court clerk and held a number of mayor's posts, so the family moved several times. Killing was already enthusiastic about his teacher for geometry at the grammar school in Brilon , where he was trained in classical languages. He began studying mathematics in the winter semester of 1865/66 in Münster , where he continued his education mainly by studying the works of Plücker, Hesse and the Disquisitiones Arithmeticae by Carl Friedrich Gauß , and continued with Ernst in Berlin for the winter semester of 1867/68 Eduard Kummer , Hermann von Helmholtz and Karl Weierstrass left . Killing became a member of the Catholic student association KDSt.V. Sauerlandia Münster in the CV . Later he also became an honorary member of the Catholic student association VKDSt. Saxonia Münster in the CV. In March 1872 Killing was in Weierstrass with a thesis The second order surface tufts on the application of elementary divisors of a matrix to the surface theory doctorate .

From 1873 to 1878 he taught at Berlin schools ( Friedrichwerdersches Gymnasium and Catholic St. Hedwig School), from 1878 at his home high school in Brilon and from 1882 as a mathematics professor at the Lyceum Hosianum in Braunsberg (recommended by Weierstrass), where he ended up being rector was. During his time as a high school teacher, he was already publishing - despite mathematical isolation and heavy demands from teaching - he taught mathematics as well as other subjects such as Latin and Greek - from 1880 onwards on non-Euclidean geometries in any dimensions. His book The non-Euclidean spatial forms in analytical treatment appeared in 1885. In the same year 1885 he was elected a member of the Leopoldina .

In 1892 he became a professor at the Westphalian Wilhelms University in Münster, where he was also rector from 1897 to 1898. Killing was a deeply religious Catholic and joined the Third Order of the Franciscans with his wife Anna at the age of 39 . He had been married since 1875 and had two daughters and four sons, who all died before him (two as children, one in 1910 while he was doing his habilitation in music history, another in 1918 during the First World War).

In his research on non-Euclidean space forms, Killing invented the Lie algebras independently of Sophus Lie (who found them around 1870 while studying differential equations) , to whose classification he wanted to trace that of the non-Euclidean space forms. He introduced the Cartan subalgebra, the Cartan matrix and the idea of ​​the root system for his classification of the ( semi- simple real) Lie algebras . He started his investigations in the yearbook of the Lyceum Hosianum of 1884. He sent a copy to Felix Klein , who first informed him about the parallel investigations by Lie. Killing also wrote to Lie, who initially did not answer and only sent him separate prints of his work on loan after he was urged again, after Killing had assured him that he was only interested in geometry and did not want to compete with him in applications to differential equations. He also came into contact with Lie's assistant, Friedrich Engel , with whom he entered into a lively correspondence from 1885. Engel encouraged him to further investigate, and in a series of four papers in the Mathematische Annalen 1888 to 1890 ( The Composition of the Continuous Finite Transformation Groups ) Killing then set out his classification. The exceptional Lie algebras, which he only published in May Discovered in 1887.

With his classification of Lie algebras he carried out one of the earliest classification programs in algebra (results on the classification of algebras and groups followed soon after in the 19th century ). His classification of Lie algebras was overshadowed for a long time by the later work of Élie Cartan (in his dissertation in 1894) , which simplified, generalized and supplemented Killings' classification, e.g. B. to explicit representations of the exceptional Lie groups, but explicitly referred to Killing in his publications. In 1900, Killing was the second after Sophus Lie to receive the Lobachevsky Prize .

The name characteristic equation of a matrix also goes back to Killing . Killing wrote a number of textbooks on geometry and elementary mathematics .

See also: killing vector field





  • AJ Coleman: The greatest mathematical paper of all time , The Mathematical Intelligencer, Vol. 11, 1989, pp. 29-38.
  • Friedrich Engel: Wilhelm Killing . In: Annual report of the German Mathematicians Association , Vol. 39, 1930, pp. 140–154 ( digitized version )
  • Thomas W. Hawkins : Wilhelm Killing and the Structure of Lie Algebras , Archive for History of Exact Science, Vol. 26, 1982, pp. 126-192.
  • Thomas W. Hawkins: Non-euclidean geometry and Weierstrassian mathematics: The background to Killing's work on Lie algebras , Historia Mathematica, Vol. 7, 1980, pp. 289-342.
  • Thomas W. Hawkins: Emergence of the theory of Lie groups , Springer 2000
  • Wolfgang Hein (editor) Wilhelm Killing: Correspondence with Friedrich Engel on the theory of Lie algebras: on the 150th birthday of Wilhelm Killing , Braunschweig, Vieweg 1997
  • Sigurdur Helgason A Centennial: Wilhelm Killing and the Exceptional Groups , The Mathematical Intelligencer, Vol. 12, 1990, pp. 54-57.
  • Otto Volk:  Killing, Wilhelm. In: New German Biography (NDB). Volume 11, Duncker & Humblot, Berlin 1977, ISBN 3-428-00192-3 , pp. 607 f. ( Digitized version ).

Web links


  1. Killing has already carried out complex studies, but was primarily interested in geometric applications.
  2. In July 1886 he also visited Lie and Engel in Leipzig.