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. is the 130th day of the year (131st in leap years) in the Gregorian calendar. ... 1847 was a common year starting on Friday (see link for calendar). ... is the 42nd day of the year in the Gregorian calendar. ... Year 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... is the 130th day of the year (131st in leap years) in the Gregorian calendar. ... 1847 was a common year starting on Friday (see link for calendar). ... is the 42nd day of the year in the Gregorian calendar. ... Year 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with 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. The University of Münster (German Westfälische Wilhelms-Universität Münster, WWU) is a public university located in the city of Münster, North Rhine-Westphalia in Germany. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Buildings of the Collegium Hosianum The Collegium Hosianum was the Jesuit collegium in Royal Prussia, Poland, founded in 1565, 1566 by Cardinal Stanislaus Hosius in Braniewo (Braunsberg). ...

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. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... Year 1880 (MDCCCLXXX) was a leap year starting on Thursday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Tuesday of the 12-day slower Julian calendar). ...

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. In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. ... In mathematics, the term Cartan matrix has two meanings. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In mathematics, a simple Lie group is a connected non-abelian Lie group G whose quotient by its center is simple as an abstract group. ...

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." In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... H.S.M. Coxeter. ...

Killing also introduced the term characteristic equation of a matrix. In linear algebra, the characteristic equation of a square matrix A is the equation in one variable λ where I is the identity matrix. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

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

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...

See also

• Killing equation
• Killing form
• Killing spinor
• Killing vector field

In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. ... Categories: Wikipedia cleanup | Stub ... In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. ...

External links

• O'Connor, John J. & Robertson, Edmund F., "Wilhelm Killing", MacTutor History of Mathematics archive 
• Wilhelm Killing at the Mathematics Genealogy Project