Wilhelm Killing

From Wikipedia, the free encyclopedia

Wilhelm Karl Joseph Killing (1847 May 10 – 1923 February 11) 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 Wilhelm Weierstrass [1] and Ernst Kummer [2] at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. He became a professor at the seminary college 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 made important contributions to the classification of simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix. Elie Cartan's [3] 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 [4] when Weyl was 3 years old and listed the orders of the Coxeter [5] transformation 19 years before Coxeter was born."

Killing also introduced the term characteristic equation of a matrix.

Source:

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.

[edit] See also

• Killing form
• Killing vector field
• Killing spinor [6]

[edit] External links

• O'Connor, John J., and Edmund F. Robertson. "Wilhelm Killing". MacTutor History of Mathematics archive.

• Mathematics Genealogy Project, [Killing http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=9715]