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Person: Killing, Wilhelm Karl Joseph

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.

Mathematical Profile (Excerpt):

• Wilhelm was brought up as a Roman Catholic and his parents gave him a conservative outlook, with a great love of his country.
• 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.
• 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.
• 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 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.
• In 1878 Killing returned to the Gymnasium in Brilon and taught at the school where he himself had been a pupil.
• 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.
• 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.
• Let us now examine in more detail how Killing's ideas on the classification developed.
• Killing introduced Lie algebras in Programmschrift Ⓣ(Manifesto) (1884) published by the Lyceum Hosianum in Braunsberg.
• 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 Ⓣ(The composition of the continuous / finite transformation groups), 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.
• discoveries were made under a number of ad hoc hypotheses to which Killing at that time could not have attached any great importance.
• It is no wonder that Killing did not publish these investigations.
• 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.
• 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.
• 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.
• Killing visited Engel and Lie in Leipzig in the summer of 1886 on his way to Heidelberg.
• 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.
• 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 GGG, and by 18 October he had discovered the complete list of simple algebras.
• Publication of the results came in the third and fourth parts of Killing's paper Die Zusammensetzung der stetigen/endlichen Transformationsgruppen Ⓣ(The composition of the continuous / finite transformation groups) referred to above.
• Finally, before we leave our discussion of Killing's work, it is worth noting that he introduced the term 'characteristic equation' of a matrix.
• 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.
• Killing was honoured with the award of the Lobachevsky Prize by the Kazan Physico-mathematical Society in 1900.
• The collapse of social cohesion in Germany after 1918 caused Killing much pain in his last years as he was a great patriot.
• 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.

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

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References

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive