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Wilhelm Karl Joseph Killing | |
---|---|

Born | 10 May 1847 |

Died | 11 February 1923 75) | (aged

Residence | Germany |

Citizenship | German |

Known for | Lie algebras, Lie groups, and non-Euclidean 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 non-Euclidean geometry.

**Germany**, officially the **Federal Republic of Germany**, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, and the Alps, Lake Constance and the High Rhine to the south. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, France to the southwest, and Luxembourg, Belgium and the Netherlands to the west.

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

In mathematics, a **Lie algebra** is a vector space together with a non-associative, alternating bilinear map , called the Lie bracket, satisfying the Jacobi identity.

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.

The **University of Münster** is a public university located in the city of Münster, North Rhine-Westphalia in Germany.

**Karl Theodor Wilhelm Weierstrass** was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany and gymnastics.

**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.

In 1878 Killing wrote on space forms in terms of non-Euclidean 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.

In mathematics, **non-Euclidean geometry** consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.

* Crelle's Journal*, or just

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 (*n*+1)-dimensional Minkowski space and *m*-planes are represented by the intersections of the (*m*+1)-planes in Minkowski space with *S*^{+}. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the *n*-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group.

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 ]}

**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.

From 1888 to 1890, Killing essentially classified the complex finite-dimensional 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.

In mathematics, a **Cartan subalgebra**, often abbreviated as **CSA**, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis.

In mathematics, the term **Cartan matrix** has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.

**Élie Joseph Cartan,** ForMemRS was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems, and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century.

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] }

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 group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

In mathematics, a **Coxeter group**, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

**Harold Scott MacDonald** "**Donald**" **Coxeter**, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

**Ferdinand Georg Frobenius** was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions, and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds.

**Elwin Bruno Christoffel** was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provide the mathematical basis for general relativity.

In mathematics, **E _{8}** is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E

**Eduard Study**, more properly **Christian Hugo Eduard Study**, was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.

**Ludwig Otto Blumenthal** was a German mathematician and professor at RWTH Aachen University.

In Lie theory and representation theory, the **Levi decomposition**, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905), states that any finite-dimensional real Lie algebra *g* is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its **radical**, a maximal solvable ideal, and the other is a semisimple subalgebra, called a **Levi subalgebra**. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra.

**Gerhard Hessenberg** was a German mathematician. He received his Ph.D. from the University of Berlin in 1899 under the guidance of Hermann Schwarz and Lazarus Fuchs. His name is usually associated with projective geometry, where he is known for proving that Desargues' theorem is a consequence of Pappus's hexagon theorem, and differential geometry where he is known for introducing the concept of a connection. He was also a set theorist: the Hessenberg sum and product of ordinals are named after him. However, Hessenberg matrices are named for Karl Hessenberg, a near relative.

**Karl Theodor Vahlen** was an Austrian-born mathematician who was an ardent supporter of the Nazi Party. He was a member of both the SA and SS.

In mathematics, **conical functions** or **Mehler functions** are functions which can be expressed in terms of Legendre functions of the first and second kind, and

**Georg Scheffers** was a German mathematician specializing in differential geometry. He was born on November 21, 1866 in the village of Altendorf near Holzminden. Scheffers began his university career at the University of Leipzig where he studied with Felix Klein and Sophus Lie. Scheffers was a coauthor with Lie for three of the earliest expressions of Lie theory:

**Friedrich Heinrich Schur** was a German mathematician who studied geometry.

In mathematics, the **Neukirch–Uchida theorem** shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Jürgen Neukirch (1969) showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Kôji Uchida (1976) strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Florian Pop extended the result to infinite fields that are finitely generated over prime fields.

**Ernst Kötter** was a German mathematician who graduated in 1884 from Berlin University. His treatise *"Fundamentals of a purely geometrical theory of algebraic plane curves"* gained the 1886 prize of the Berlin Royal Academy. In 1901, he published his report on *"The development of synthetic geometry from Monge to Staudt (1847)"*; it had been sent to the press as early as 1897, but completion was deferred by Kötter's appointment to Aachen University and a subsequent persisting illness. He constructed a mobile wood model to illustrate the theorems of Dandelin spheres.

**Reinhardt Kiehl** is a German mathematician.

**Ioannis "John" N. Hazzidakis** was a Greek mathematician, known for the Hazzidakis transform in differential geometry. The Hazzidakis formula for the Hazzidakis transform can be applied in proving Hilbert's theorem on negative curvature, stating that hyperbolic geometry does not have a model in 3-dimensional Euclidean space.

**Eduard Wirsing** is a German mathematician, specializing in number theory.

**Carl Friedrich Geiser** was a Swiss mathematician, specializing in algebraic geometry. He is known for the Geiser involution and Geiser's minimal surface.

**Friedrich Bachmann** was a German mathematician who specialised in geometry and group theory.

- ↑ Hawkins, Thomas,
*Emergence of the Theory of Lie Groups,*New York: Springer, 2000. - ↑ Reynolds, W. F. (1993). "Hyperbolic geometry on a hyperboloid".
*The American Mathematical Monthly*.**100**(5): 442–455. JSTOR 2324297. - ↑ Ratcliffe, J. G. (1994). "Hyperbolic geometry".
*Foundations of Hyperbolic Manifolds*. New York. pp. 56–104. ISBN 038794348X. - ↑ Coleman, A. John, "The Greatest Mathematical Paper of All Time,"
*The Mathematical Intelligencer,*vol. 11, no. 3, pp. 29–38.

- Work on non-Euclidean 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 Nicht-Euklidischen Raumformen".
*Journal für die reine und angewandte Mathematik*.**89**: 265–287. - Killing, W. (1885) [1884]. "Die Mechanik in den Nicht-Euklidischen Raumformen".
*Journal für die reine und angewandte Mathematik*.**98**: 1–48. - Killing, W. (1885).
*Die nicht-euklidischen Raumformen*. Leipzig: Teubner. - Killing, W. (1891). "Ueber die Clifford-Klein'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.

- O'Connor, John J.; Robertson, Edmund F., "Wilhelm Killing",
*MacTutor History of Mathematics archive*, University of St Andrews .

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