w.killing komplett

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Wilhelm Killing (1847-1923)

Wilhelm Carl Joseph Killing was born 10 February 1847 in Burbach, a smalltown near Siegen. His father Joseph was a legal clerk who had married the

daughter of the local chemist, named Kortenbach. They had three children

Wilhelm, Hedwig and Karl.

In his autobiographical notes Killing describes himself as a child being

frail and besides very awkward, the perpetually excited, but completely

impractical bookworm. His family was Roman Catholic and of conservative

convictions. The father Joseph, who subsequently became mayor of various small

towns of the region, evinced a high sense of duty. This subsequently was to

become also a distinguishing feature of his son Wilhelm.

Killing attended elementary school at various places but was also privately

prepared by local clergymen to enter the Gymnasium in Brilon. He was much

attracted by classical Latin, Greek and Hebrew. Only somewhat later awoke his

love of mathematics, in particular geometry; stimulated and fostered by his teacher

Harnischmacher who encouraged him to study Eulers Introductio. Wilhelm was

doing so even shortly before his school-leaving examination (Abitur) and so

somewhat neglected the preparations for the exam (at least he said so in his

autobiographical notes, but his examination marks turned out to be excellent).

Killing was later to express his reverence for Harnischmacher by dedicating his

dissertation to him.

Killing decided to study mathematics, and in the autumn of 1865 he started a

course of study at Mnster. The University of Mnster opened in 1780 but after the

Napoleonic wars it lost its status of a full university and was reduced to a college of

limited rights and obligations. Since 1843 it had continued under the name of a

Royal Academy.

When Killing came to this academy he met a situation far less favourable to

him than it had been to Weierstrass 27 years ago. The standard which Gudermann

had brought to the academy, Gudermanns successor could not maintain. So Killing

had to stand very much on his own; he particularly studied the work of Plcker on


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geometry, some of it very meticulously (as he later was to say). After four terms

Killing moved to the University of Berlin. Here he found the highest quality of

teaching, represented by Kummer and Weierstrass. The greatest stimulus for him

was Weierstrasss Seminar, where many new ideas were presented, mostly byWeierstrass himself, and in various fields of mathematics, including algebra and

geometry. But in 1870 Killing interrupted his studies, when his father, at that time

mayor of the little town Rthen, pleaded with him to teach at the local school,

which was then in a precarious state. In 1871 Killing returned to Berlin and soon

began working towards his doctorate, supervised by Weierstrass. He presented his

dissertation in March 1872. Its title was Der Flchenbschel zweiter Ordnung

(Bundles of surfaces of the second order). A geometric theme but in substance very

algebraic, in applying Weierstrasss theory around his elementary divisors, which

by the way also included what we now call the Jordan canonical form of matrices.

In the autumn of the same year Killing obtained the provisional teaching

qualification in mathematics and physics for grammar school tuition, and also Greek

and Latin (at lower levels). After he had gone through the customary probation time

he taught at various schools in Berlin, among them St. Hedwigs Catholic school and

the Friedrichwerdersche Gymnasium (at that time one of the most renowned schools

within Prussia. (One of its teachers was Du Bois-Reymond, and among its pupils you

find names like Ludwig Thieck, Karl Gutzkow, and in a later time Max Liebermann

and Victor Klemperer.)

In 1875 Killing married Anna Commer, daughter of the musicologist and

composer Franz Commer. Wilhelm and Anna Killing were to have seven children.

Two sons and a daughter died in infancy, the third son shortly before the completion

of his habilitation thesis in musicology. The fourth son fell ill in a military camp and

died shortly before the end of First World War in 1918. Only two daughters outlived

their parents.

In 1878 Killing returned to Brilon to teach at the same Gymnasium where he

himself had been a pupil. There he acquired also an additional qualification for

religious instruction. Despite of his heavy teaching load he published in 1879 his first

paper (On two space forms with constant positive curvature) in Crelles Journal. Two

further papers by him, on non-euclidean geometry in n dimensions, also appeared in


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Crelles Journal (1880 and 1885). Killing there joined the current research of Non-

Euclidean Geometry, a subject having been newly stimulated by what was found in

Gausss unpublished papers and of course by the spread of Riemanns ideas. Killing

had a particular devotion to the axiomatics of geometry. His examination ofgeometrys foundation seem to have been influenced by Helmholtz, the great

physicist, who, in 1868 in Berlin had presented his paper ber die Tatsachen, die

der Geometrie zum Grunde liegen (On the facts upon which geometry is founded).

In another line of papers Killing was to use the impetus of certain ideas of

Riemann and Weierstrass to start a programmatic project of his own. These papers

appeared in the Programme of the Gymnasium Brilon, and later on in the Programme

of the Academy in Braunsberg (todays Braniewo). With this we already entered the

most important period of Killings life.

It was on a recommendation by Weierstrass that Killing was appointed to the

vacant chair of mathematics in 1882.

At this moment I wish to say a few words regarding the history of Braunsbergs

Academy, also called the Lyceum Hosianum.

In 1665 cardinal Stanislaus Hosius, bishop of Warmia (Ermland in German),

initiated the foundation of a Jesuit College at Braunsberg, supplemented by a

seminary for boys. Its purpose was to counter the widespread Protestant movement in

Prussia (and elsewhere). Warmia at that time was under the Polish Crown (though

enjoying substantial autonomy); most of its population was German speaking (in fact

a peculiar dialect of German), and it was predominantly Catholic (in contrast to rest

of the population in the province of Royal Prussia).

Both institutions, the College and the Gymnasium (as we may call the boyss

seminary) thrived. The Jesuit College taught, in addition to standard subjects,

mathematics, dialectic and German language and literature. It even attracted students

from the northern countries and various other parts of Europe.

A first crisis was caused by the Swedish occupation of Braunsberg from 1626 to

1635. The College was closed down but once the Swedes had left, the Jesuits

returned the very next day. Their precious library had been shipped to Sweden (and

there it remains, still in the possession of the University of Upsala). The Jesuits had to


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begin anew and reorganized their institutions. The College regained and upheld a

high standard, and in 1702, justifiably, a letter was sent to Rome petitioning the Pope

to elevate the College to a University. But all such hopes were ruined by the Nordic

War (1700-1721), when Braunsberg suffered from repeated occupations by German,Swedish and Polish troops.

In 1772 Warmia lost its autonomy and passed to the Crown of Prussia. One year

later the Pope in Rome abolished the Order of Jesuits. King Frederick II of Prussia

interdicted the announcement of the papal bull in his country and decreed that the ex-

Jesuits should continue with their teaching. But this clash was only to bring about the

end of the College. In 1811 its activities were terminated. The Prussian government

at first thought of moving the training of Catholic priests to the University of

Knigsberg, but this was strongly opposed by Varmias then bishop Wilhelm Prince

of Hohenzollern. He won the argument and in 1818 the Lyceum Hosianum was

founded as successor to the Jesuit College, starting tuition in 1821.

The former Jesuit Gymnasium, that is the initial seminary school, had in 1781

been renamed Academic Gymnasium, under which it continued to eke out a

miserable existence. But in 1811 it started a fresh life under the name of Royal

Catholic Gymnasium. Its first Headmaster, Heinrich Schmlling, had come from

Mnsters Gymnasium Paulinum. Later he simultaneously was professor at the

Lyceum Hosianum as well. The Royal Catholic Gymnasium of Braunsberg

prospered; it was enjoying an excellent reputation when Weierstrass came to teach

there.

After this little excursion let me come back to Killing.

In Braunsberg Killing had to teach a wide range of topics (among them popular

astronomy and basic chemistry). Mathematically he was rather isolated, lacking also a

library current in his subject. Soon he became Rector of the Academy (probably

before 1886), and additionally he was engaged in other affairs of public life. But

despite that he was working hard on his research (How did he manage all that, one

wonders.) In 1884 he presented what is called his Programmschrift entitled The

Extension of the Concept of space. As I said, it appeared in the Programme of the


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Academy in Braunsberg, but the name Programmschrift serves well to describe a

grand programmatic project.

In this paper Killing introduced what we now call Lie algebras. His aim was to

study all space forms that is geometries with certain properties relating toinfinitesimal motions. This led him to the project of classifying all finite-dimensional

Lie algebras (By that was asking for the Impossible. Better to stick to the simple Lie

algebras, which he soon did. Furthermore he realized that passing on to Lie algebras

over the complex numbers would be of great help.)

Portrait of Killing

Lie algebras, in fact, were first invented by Sophus Lie, while seeking a new

approach to differential equations. (Whereas Killing about ten years later

independently discovered Lie algebras in his attempt to classify all types of his space

forms.) In Braunsberg Killing had no access to the journal in which Lies papers

appeared (with little resonance for some time). Killing sent a copy of his

Programmschrift to Felix Klein, and Klein told him that his concepts were closely

related to the structures that Lie was interested in, and that Lie had published a

number of papers on them over the preceding ten years.

In October 1885 Killing wrote to Lie asking for copies of Lies papers and

assuring him that his interest in Lie algebras was limited to geometrical

considerations. After Klein had told Killing that in Christiania Engel was working on

his habiliation on transformation groups under Lie, Killing wrote to Engel, and thus

began a correspondence of many years which was of great benefit to Killing (and to

Engel as well).

In August of 1886 Killing, in his capacity as rector, had to represent the Lyceum

Hosianum at the University of Heidelberg. On his journey he visited Engel and Lie at

Leipzig, where Lie was professor and Engel held a lectureship. But Lie, then already

quite a famous man, was still resentful of Killing, and a workable relationship

between the two men did not arise, neither at their meeting at Leipzig nor in the years

to come.


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The more important for Killing was the great help and encouragement he got

from Friedrich Engel. But also another good angel stood at his side, and a mighty one

at that, and this was the new algebra he had learnt from Weierstrass. With that he

undertook - what even today would seem too much for a single man - Theclassification of simple Lie algebras (over the complex numbers). His results were

published in his paper Die Zusammensetzung der stetigen endlichen

Transformationsgruppen, of which parts I and II appeared already in 1888, followed

by parts III and IV in the two years thereafter (altogether nearly 200 pages in the

Mathematische Annalen).

What Killing had established was a complete list of all simple Lie algebras

(over C), consisting of four infinite series (corresponding to the classical groups),

and a finite set of exceptional algebras (exactly five in number). A work of gigantic

stature. Not likely that such a paper was immediately to meet general

acknowledgement, let alone full appreciation. In fact, it took almost a century before

Killing got the common praise he deserved. Why only after so much time?

Because mathematicians in general have only little or no interest in history.

They go for the ready-made, not bothering to asking where it originally comes from.

Eli Cartans dissertation of 1894, (a true masterpiece, by the way,) is a case in point.

As it is sometimes better not to try and express in ones own words what others

already phrased to perfection, I shall quote John Coleman: Cartan did give a

remarkably elegant and clear exposition of Killings result. He also made an essential

contribution.by proving that the Cartan subalgebra of a simple Lie algebra is

abelian. This property was announced by Killing but his proof was invalid. In parts,

other than part II, of Killings paper there are major deficiencies which Cartan

corrected, notably in the treatment of nilpotent Lie algebras. In the last third of

Cartans thesis, many new and important results are based upon and go beyond

Killings work. Coleman also pointed out: Cartan was meticulous in noting his

indebtedness to Killing. In Cartans thesis there are 20 references to Lie and 63 to

Killing.

In any case, it was Cartans paper mathematicians subsequently would refer to,

as did Hermann Weyl 1926 in his important article on the representations of semi-

simple groups. So, for a long time, Killings work had been overshadowed by the


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work of Cartan. The reason why it was neglected can hardly be better expressed than

by Helgason when he writes: I think that in modern terminology it is fair to say that

Cartans thesis represented a friendly takeover.

On the other hand, everyone looking at Cartans thesis with the least sense ofhistory would not fail to register the name of Killing, and finally this name has

become famous among mathematicians worldwide. The final breakthrough perhaps

was done by the articles of John Coleman and Sigurdur Helgason dedicated to the

centennial of Killings paper of 1888.

Helgason said that Killings classification of the simple Lie algebras over C

turned out to be a milestone in the history of mathematics and that it furnished the

impetus towards the problem of classifying finite simple groups. In particular he

writes: The exceptional simple Lie algebras are the subject of the final 18 [the

final section] in Killings paper. This is certainly his most remarkable discovery,

although these algebras appeared to him at first a kind of nuisance, which he tried to

eliminate ... they have subsequently played important roles in Lie theory

Indeed they were used in the construction of the sporadic simple finite groups.

And as Coleman points out, the largest of them is now the darling of super-string

theorists.

After Killing had published his paper in the Mathematische Annalen, he did not

stay without recognition in his time. He was offered a chair of mathematics at

Mnsters Academy, which he was appointed to in 1892.

Killing was to spend the rest of his life at Mnster, deeply devoted to his

teaching. For that he also wrote several books on geometry, one of which has gone

through repeated editions right up to the present. He also published a series of smaller

research papers, not to be compared with his giant work of 1888.

Killing was also very much engaged in the public affairs of the University and

his Church. For example, he was the president of a charitable society for ten years.

Killing was loved and admired by his students. For them he worked unsparingly

of time and energy. But he also demanded from them, particularly in his Seminar, that

they work with enthusiasm and true dedication to their subject. Weierstrass was his

model, and when Killing in 1896 became rector of the Academy his inaugural speech

was on Karl Weierstrass. (It was published and indeed is still very readable.)


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Expecting that once it will turn out.

But suddenly he sees that the useless itself becomes useful

When the very Greatest is resting upon the very Smallest.

w.killing komplett

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