Wilhelm Killing
Encyclopedia
Wilhelm Karl Joseph Killing (May 10, 1847– February 11, 1923) was a German
mathematician
who made important contributions to the theories of Lie algebra
s, Lie group
s, 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 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.
In 1878 Killing wrote on space forms in Crelle's Journal
86:72–83. Two years later he wrote on computations in hyperbolic geometry
in the same journal. Recounting lectures of Weierstrass, he there introduced the hyperboloid model
described by Weierstrass coordinates.
Killing invented Lie algebra
s 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.
From 1888 to 1890, Killing essentially classified the complex finite dimensional simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix
. Élie Cartan
's 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 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."
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who made important contributions to the theories of Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s, Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
.
Killing studied at the University of Münster
University of Münster
The University of Münster is a public university located in the city of Münster, North Rhine-Westphalia in Germany. The WWU is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities...
and later wrote his dissertation under Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
and Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life: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
Collegium Hosianum
The Collegium Hosianum was the Jesuit collegium in Royal Prussia, Poland, founded in 1565, 1566 by Cardinal Stanislaus Hosius in Braniewo . The city of Braniewo was then part of the Diocese of Warmia , a semi-autonomous part of Poland with cardinal Hosius as its prince-bishop...
in Braunsberg (now Braniewo
Braniewo
Braniewo is a town in northeastern Poland, in the Warmian-Masurian Voivodeship, with a population of 18,068 . It is the capital of Braniewo County...
). 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.
In 1878 Killing wrote on space forms in Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
86:72–83. Two years later he wrote on computations in hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
in the same journal. Recounting lectures of Weierstrass, he there introduced the hyperboloid model
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model , is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are...
described by Weierstrass coordinates.
Killing invented Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s independently of Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
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.
From 1888 to 1890, Killing essentially classified the complex finite dimensional simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix
Cartan matrix
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.- Lie algebras :A generalized...
. Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
's dissertation was essentially a rigorous rewriting of Killing's paper. Killing also introduced the notion of a root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
. He discovered 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
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."
See also
- Killing equation
- Killing formKilling formIn mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...
- Killing horizonKilling horizonA Killing horizon is a null hypersurface on which there is a null Killing vector field .Associated to a Killing horizon is a geometrical quantity known as surface gravity, \kappa...
- Killing spinorKilling spinorKilling spinor is a term used in mathematics and physics. By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistorspinors which are also eigenspinors of the Dirac operator, see Th. Friedrich ,...
- Killing tensor
- Killing vector fieldKilling vector fieldIn mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...