# Wilhelm Karl Joseph Killing

### Quick Info

Born
10 May 1847
Burbach (near Siegen), Westphalia, Germany
Died
11 February 1923
Münster, Germany

Summary
Wilhelm Killing was a German mathematician who introduced Lie algebras independently of Lie in his study of non-euclidean geometry. His classification of the simple Lie algebras was one of the finest achievements in the whole of mathematical research.

### Biography

Wilhelm Killing's mother was Anna Catharina Kortenbach and his father was Josef Killing. Josef was trained as a legal clerk and his first job was in Burbach about 15 km south of Siegen. There he married Catharina Kortenbach, the daughter of the pharmacist Wilhelm Kortenbach. Wilhelm Killing, the subject of this biography, was one of their three children, the other two being Hedwig and Karl. When Wilhelm was three years old the family moved to Medebach which is about 70 km north east of Siegen. As a child Wilhelm's health was not good and he was described as:-
... quite weakly and besides very awkward ..., always excited, but a completely unpractical bookworm.
Wilhelm was brought up as a Roman Catholic and his parents gave him a conservative outlook, with a great love of his country. After ten years in Medebach the family moved again, this time to Winterberg which is less than 15 km west of Medebach. Josef Killing was mayor of Medebach, then of Winterberg and, in 1862 he became mayor of Rüthen which is about 30 km north of Winterberg.

Killing attended elementary school and was also given private tutoring by local clergymen to prepare him to enter the Gymnasium in Brilon. The first subjects to attract Killing at the Gymnasium were the classical languages of Greek, Latin and Hebrew. It was his teacher Harnischmacher who first gave Killing his love of mathematics; later he expressed his admiration for Harnischmacher when he dedicated his thesis to him. In particularly the study of geometry at the Gymnasium convinced Killing that he should become a mathematician. He graduated from the Gymnasium in 1865 and in the autumn of the same year began his university studies at Münster. The Westphalian Wilhelm University of Münster was founded 1780, but only became a full university in 1902. When Killing studied there it was a Royal Academy. The lecturer in mathematics and astronomy at the Academy was Eduard Heis but he did not teach mathematics to a high level and Killing learnt his mathematics from studying books on his own: in particular he read Plücker's works on geometry and tried to extend the results which Plücker proved. He also read works by Hesse and he read Gauss's Disquisitiones Arithmeticae.

At Münster Killing was having to educate himself, and although he greatly appreciated the genius of the authors whose works he read, he felt that without expert teaching he was not getting as much out of his studies as he should. After four terms he moved to Berlin, matriculating there for the winter semester 1867-68. At Berlin, unlike Münster, he found the highest quality of teaching and he was particularly influenced by Kummer, Weierstrass and Helmholtz. He interrupted his studies in 1870-71 when his father asked him to return to help at the school in Rüthen. He returned to his studies at the University of Berlin in 1871 and soon began work towards his doctorate supervised by Weierstrass. His doctoral thesis, which applied Weierstrass's theory of elementary divisors of a matrix to surfaces, was presented in March 1872. It was entitled Der Flächenbüschel zweiter Ordnung .

After completing his doctorate Killing trained to become a Gymnasium teacher of mathematics and physics, also qualifying to teach Greek and Latin at a lower level. He qualified in 1873 and spent a year as a probationary teacher. Until 1878 he taught at schools in Berlin; the Frdr Werder Gymnasium and St Hedwig's Catholic school. In 1875 he married Anna Commer, daughter of a lecturer in music. They had four sons, the first two of whom died as infants, and two daughters Maria and Anka. In 1878 Killing returned to the Gymnasium in Brilon and taught at the school where he himself had been a pupil. He had a heavy teaching load and during much of this time he would spend around 36 hours each week either teaching in the classroom or tutoring pupils. Despite this he published his first paper Über zwei Raumformen mit konstanter positiver Krümmung in 1879 in Crelle's Journal and two further papers, also in Crelle's Journal, on non-euclidean geometry in $n$-dimensions: Die Rechnung in den Nicht-Euklidischen Raumformen (1880) and Die Mechanik in den Nicht-Euklidischen Raumformeni (1885). He published the book Die nichteuklidischen Raumformen in analytischer Behandlung on non-euclidean geometry in Leipzig in 1885.

On Weierstrass's recommendation Killing was appointed to a chair of mathematics at the Lyceum Hosianum in Braunsberg in 1882. Killing spent ten years in Braunsberg, isolated mathematically, but during this period he produced some of the most original mathematics ever produced. Lie algebras were introduced by Lie in about 1870 in his work on differential equations. Killing introduced them independently with quite a different purpose since his interest was in non-euclidean geometry. The classification of the semisimple Lie algebras by Killing was one of the finest achievements in the whole of mathematical research. The main tools in the classification of the semisimple Lie algebras are Cartan subalgebras and the Cartan matrix both first introduced by Killing. He also introduced the idea of a root system which appears throughout much of the algebra of today. Let us now examine in more detail how Killing's ideas on the classification developed.

Killing introduced Lie algebras in Programmschrift (1884) published by the Lyceum Hosianum in Braunsberg. His aim was to systematically study all space forms, that is geometries with specific properties relating to infinitesimal motions. In his Programmschrift he translated this geometrical aim into the problem of classifying all finite dimensional real Lie algebras. At this stage Killing was not aware of Lie's work and therefore his definition of a Lie algebra was made quite independently of Lie. Although the classification theorems were presented by Killing in his paper Die Zusammensetzung der stetigen/endlichen Transformationsgruppen , which was published in four parts in Mathematische Annalen between 1888 and 1890, it is clear that when he published Programmschrift he already had the main ideas in place of how the classification would proceed. We should make it clear that although he was examining conditions on a Lie algebra which essentially made it semisimple (that is having no soluble ideals) in Programmschrift, he was not aiming at such a classification at this stage. Rather he was examining conditions on the Lie algebra which he studied for their geometrical significance and only later did he try to relate the conditions to semisimple algebras. Hawkins writes that Killing's [3]:-
... discoveries were made under a number of ad hoc hypotheses to which Killing at that time could not have attached any great importance. Furthermore, he had permitted complex numbers into the calculations to facilitate the analysis, but eventually, for his classification of space forms, he must deal with the "real" case. It is no wonder that Killing did not publish these investigations. They were far too inconclusive for public exposure, even in the form of a Programmschrift. It would have been more reasonable for him to have abandoned his attempt to classify space forms since he had at least pursued the problem far enough to realise just how formidable it was.
Killing sent Klein a copy of Programmschrift in July 1884 and Klein replied by telling him that what he was looking at was closely related to structures that Sophus Lie was interested in, and that Lie had published a number of papers on these algebras over the preceding ten years. Killing responded by sending a copy of Programmschrift to Lie in August 1884. On receiving no reply he wrote again to Klein who told him that Engel was working in Christiania on his habilitation on transformation groups under Lie. In October 1885 Killing wrote again to Lie, this time requesting copies of Lie's papers and assuring him that his interest in Lie algebras was limited to geometrical considerations. Lie sent copies of his papers to Killing who considered that he only had them on loan and had to return them, which he did in around March 1886. He had not had time to fully appreciate all that they contained. However Killing had also written to Engel in November 1885 and they started a long scientific correspondence which was helpful to them both.

It is fair to say that without the encouragement and interest shown by Engel, Killing might not have pushed forward with his work on Lie algebras. They discussed the simple Lie algebras which they knew about and Killing conjectured (wrongly) on 12 April 1886 that the only simple algebras were those related to the special linear group and orthogonal groups. In the same letter he conjectured other theorems about Lie algebras. Hawkins writes [3]:-
It is not difficult to imagine the amazement with which Engel read Killing's letter with its bold conjectures. here was an obscure professor at a Lyceum dedicated to the training of clergymen in the far-away reaches of east Prussia, discoursing with authority and conjecturing profound theorems on Lie's theory of transformation groups, a theory which had seemed an area of mathematics known to relatively few mathematicians and mastered by even fewer.
Killing visited Engel and Lie in Leipzig in the summer of 1886 on his way to Heidelberg. At this time Killing was rector of the Lyceum Hosianum in Braunsberg and in this capacity he was visiting its sister institution in Heidelberg. He arrived in Leipzig, where Lie was the professor and Engel was a dozent, on 31 July. It was not a particularly fruitful visit for, although the three men should have had a wealth of mathematical ideas to discuss, there seems to have been a personality clash between Killing and Lie. While in Leipzig, Killing also met Schur and Study. Moving on to spend August in Heidelberg, Killing did little further mathematics that year since he became concerned for the health of one of his daughters after his return to Braunsberg.

When Killing wrote to Engel on 27 April 1887 he had come up with the definition of a semisimple Lie algebra (his definition that such an algebra had no abelian ideals is equivalent to the definition that such an algebra has no soluble ideals). By the time he wrote to Engel on 23 May Killing had discovered that his conjecture about simple algebras was wrong, for he had discovered $G$, and by 18 October he had discovered the complete list of simple algebras. However, he did not have concrete representations of these algebras. Publication of the results came in the third and fourth parts of Killing's paper Die Zusammensetzung der stetigen/endlichen Transformationsgruppen referred to above. The most remarkable part of this work is his discovery of the exceptional simple Lie algebras. Helgason writes [5]:-
The exceptional simple Lie algebras are the subject of the final Section 18 of Killing's paper. This is certainly his most remarkable discovery, although these algebras appeared to him at first as a kind of nuisance, which he tried hard to eliminate. .. they have subsequently played important roles in Lie theory ...
Finally, before we leave our discussion of Killing's work, it is worth noting that he introduced the term 'characteristic equation' of a matrix.

It was Cartan, in his doctoral thesis submitted in 1894, who found concrete representations of all the exceptional simple Lie algebras (although he did not work out all the details in his thesis). He also reworked Killing's proofs to make them more easily understood. In many ways Cartan was so successful in presenting Killing's classification of the semisimple Lie algebras in rigorous and complete single work, that Killing has not received as much acclaim for his remarkable achievements as one might have expected.

We return to a description of the final stage of Killing's career. In 1892 he returned to Münster as professor of mathematics and he spent the rest of his life there submerged in teaching, administration and charitable work. He was rector of the University of Münster in 1897-98. He always upheld tradition and disliked change. One example of this was his desire that philosophy be retained as compulsory for all graduate students. He fought vigorously to retain the philosophy examination although, as Engel stated:-
Killing could not see that for most candidates the test in philosophy was completely worthless.
Killing was honoured with the award of the Lobachevsky Prize by the Kazan Physico-mathematical Society in 1900. This was the second award made of the Prize, the first in 1897 going to Lie.

The collapse of social cohesion in Germany after 1918 caused Killing much pain in his last years as he was a great patriot. He had already suffered the loss of two infant sons, but even more devastating was the loss of his remaining two sons, one of whom died in 1910 while working for his habilitation on a topic on the history of music, the other became ill in an army camp and died shortly before end of World War I in 1918.

Coleman writes in [1]:-
Throughout his life Killing evinced a high sense of duty and a deep concern for anyone in physical or spiritual need. He was steeped in what the mathematician Engel characterised as "the rigorous Westphalian Catholicism of the 1850s and 1860s". St Francis of Assisi was his model, so at the age of 39 he, together with his wife, entered the Third Order of the Franciscans. His students loved and admired Killing because he gave himself unsparingly of time and energy to them, never being satisfied for them to become narrow specialists, so he spread his lectures over many topics beyond geometry and groups.
This makes Killing look almost a mathematical saint, but this probably goes too far. He certainly lacked a sense of humour and he [3]:
... was extremely sensitive to criticism.

### References (show)

1. A J Coleman, The greatest mathematical paper of all time, The Mathematical Intelligencer 11 (1989), 29-38.
2. F Engel, Wilhelm Killing, Jahresberichte der Deutschen Mathematiker vereinigung 39 (1930), 140-154.
3. T Hawkins, Wilhelm Killing and the Structure of Lie Algebras, Archive for History of Exact Science 26 (1982), 126-192.
4. T Hawkins, Non-euclidean geometry and Weierstrassian mathematics : The background to Killing's work on Lie algebras, Historia Mathematica 7 (1980), 289-342.
5. S Helgason, A Centennial : Wihelm Killing and the Exceptional Groups, The Mathematical Intelligencer 12 (1990), 54-57.

### Additional Resources (show)

Other websites about Wilhelm Killing:

Written by J J O'Connor and E F Robertson
Last Update February 2005