威廉·基灵Wilhelm Killing

威廉·基灵的母亲是安娜·卡塔琳娜·科滕巴赫，父亲是Josef Killing。约瑟夫接受过法律文书的培训，他的第一份工作是在锡根以南约15公里的布尔巴赫。在那里，他与药剂师基灵科滕巴赫的女儿卡塔琳娜·科滕巴赫结婚。本传记的主人公基灵是他们的三个孩子之一，另外两个是海德维希和卡尔。当基灵三岁时，全家搬到了锡根东北约70公里的梅德巴赫。小时候基灵的身体不好，他被描述为：-

……相当虚弱，而且非常笨拙……，总是兴奋，但完全是个不切实际的书呆子。

基灵从小被培养为罗马天主教徒，他的父母给了他保守的观点，以及对国家的热爱。在梅德巴赫住了十年后，全家再次搬家，这次搬到了梅德巴赫以西不到15公里的温特贝格。Josef Killing曾是梅德巴赫的市长，然后是温特贝格的市长，1862年他成为温特贝格以北约30公里的吕滕的市长。

基灵上过小学，还接受当地神职人员的私人辅导，为进入Brilon的文理中学做准备。在文理中学，最早吸引基灵的科目是希腊语、拉丁语和希伯来语等古典语言。是他的老师Harnischmacher首先让基灵爱上了数学；后来，当他把学位论文献给Harnischmacher时，表达了对他的钦佩。特别是在文理中学学习几何使基灵确信自己应该成为一名数学家。他于1865年从文理中学毕业，同年秋天开始在Münster上大学。威斯特伐利亚的基灵大学Münster建于1780年，但直到1902年才成为一所完整的大学。当基灵在那里学习时，它是一所皇家科学院。该科学院的数学和天文学讲师是Eduard Heis，但他不教授高水平的数学，基灵通过自学书籍学习数学：特别是他阅读了尤里乌斯·普吕克的几何著作，并试图扩展尤里乌斯·普吕克证明的结果。他还阅读了奥托·黑塞的著作，并阅读了卡尔·弗里德里希·高斯的Disquisitiones Arithmeticae。

基灵不得不自学，尽管他非常欣赏所读著作作者的天才，但他觉得没有专家的教导，他的学习效果不如应有的那样好。四个学期后，他转到柏林，在1867-68年冬季学期注册入学。在柏林，与明斯特不同，他找到了最高质量的教学，尤其受到恩斯特·爱德华·库默尔、卡尔·魏尔斯特拉斯和赫尔曼·冯·亥姆霍兹的影响。1870-71年，他中断学业，因为父亲要求他回吕滕的学校帮忙。1871年，他回到柏林大学继续学业，并很快开始在卡尔·魏尔斯特拉斯的指导下攻读博士学位。他的博士论文将卡尔·魏尔斯特拉斯的矩阵初等因子理论应用于曲面，于1872年3月提交。论文题为Der Flächenbüschel zweiter Ordnung Ⓣ（二阶曲面束）。

完成博士学位后，基灵接受培训成为文理中学的数学和物理教师，同时也取得了在较低级别教授希腊语和拉丁语的资格。他于1873年获得资格，并担任了一年的试用教师。直到1878年，他在柏林的学校任教：弗尔德尔文理中学和圣海德维希天主教学校。1875年，他与音乐讲师之女安娜·科默结婚。他们有四个儿子，前两个在婴儿期夭折，还有两个女儿玛丽亚和安卡。1878年，基灵回到布里隆的文理中学，在他自己曾就读的学校任教。他的教学负担很重，在这段时间的大部分时间里，他每周要么在课堂上教学，要么辅导学生，大约要花36个小时。尽管如此，他于1879年在Crelle's Journal上发表了他的第一篇论文Über zwei Raumformen mit konstanter positiver Krümmung Ⓣ（论具有常正曲率的二维形状），并在Crelle's Journal上又发表了两篇关于non-euclidean geometry在$n$维中的论文：Die Rechnung in den Nicht-Euklidischen Raumformen Ⓣ（非欧空间形式中的陈述）（1880年）和Die Mechanik in den Nicht-Euklidischen Raumformeni Ⓣ（非欧空间形式中的力学）（1885年）。他于1885年在莱比锡出版了关于非欧几何的书Die nichteuklidischen Raumformen in analytischer Behandlung Ⓣ（非欧空间形式的分析处理）。

在卡尔·魏尔斯特拉斯的推荐下，基灵于1882年被任命为布劳恩斯贝格霍西亚努姆学园的数学讲席。基灵在布劳恩斯贝格度过了十年，在数学上孤立无援，但在此期间，他产生了一些最具原创性的数学成果。李代数由索菲斯·李在大约1870年在他关于微分方程的工作中引入。基灵独立地引入了它们，目的完全不同，因为他的兴趣在于非欧几何。基灵对半单李代数的分类是整个数学研究中最杰出的成就之一。半单李代数分类的主要工具是埃利·嘉当子代数和埃利·嘉当矩阵，两者均由基灵首次引入。他还引入了根系统的概念，这一概念在当今许多代数中都有出现。现在让我们更详细地考察基灵关于分类的思想是如何发展的。

基灵在ProgrammschriftⓉ（宣言）（1884年）中引入了李代数，该文由布劳恩斯贝格的霍西亚努姆中学出版。他的目标是系统地研究所有空间形式，即具有与无穷小运动相关的特定性质的几何。在他的Programmschrift中，他将这一几何目标转化为对所有有限维实李代数进行分类的问题。在这个阶段，基灵并不知道索菲斯·李的工作，因此他对索菲斯·李代数的定义是完全独立于索菲斯·李的。尽管分类定理由基灵在其论文Die Zusammensetzung der stetigen/endlichen TransformationsgruppenⓉ（连续/有限变换群的合成）中提出，该论文于1888年至1890年间在Mathematische Annalen分四部分发表，但很明显，当他发表Programmschrift时，他已经有了分类将如何进行的主要想法。我们应该明确，尽管他在Programmschrift中考察了李代数上本质上使其成为半单（即没有可解理想）的条件，但在这个阶段他并不旨在进行这样的分类。相反，他考察李代数上的条件是为了研究其几何意义，只是后来他才试图将这些条件与半单代数联系起来。霍金斯写道，基灵的[3]：-

……这些发现是在若干当时基灵不可能赋予太大重要性的特设假设下作出的。此外，他允许复数进入计算以便于分析，但最终，为了他对空间形式的分类，他必须处理“实”的情形。难怪基灵没有发表这些研究。它们太不确定，不宜公之于众，即使是以Programmschrift的形式。对他来说，更合理的做法是放弃对空间形式进行分类的尝试，因为他至少已经把问题研究得足够深入，从而意识到它是多么艰巨。

基灵于1884年7月将一份Programmschrift寄给菲利克斯·克莱因，菲利克斯·克莱因回信告诉他，他所研究的东西与索菲斯·李感兴趣的结构密切相关，并且索菲斯·李在过去十年里已经发表了若干关于这些代数的论文。基灵于1884年8月将一份Programmschrift寄给索菲斯·李作为回应。由于没有收到回复，他又写信给菲利克斯·克莱因，后者告诉他Engel正在克里斯蒂安尼亚跟随索菲斯·李研究他的关于变换群的habilitation。1885年10月，基灵再次写信给索菲斯·李，这次是索取索菲斯·李论文的副本，并向他保证自己对李代数的兴趣仅限于几何方面的考虑。索菲斯·李将自己的论文副本寄给基灵，后者认为自己只是借阅，必须归还，他大约在1886年3月归还了。他没有时间充分领会其中所包含的全部内容。然而，基灵也在1885年11月写信给Engel，他们开始了长期的科学通信，这对双方都有帮助。

可以公平地说，如果没有Engel所表现出的鼓励和兴趣，基灵可能不会推进他在李代数方面的工作。他们讨论了他们所知道的单李代数，基灵于1886年4月12日（错误地）猜想，唯一的单代数是与特殊线性群和正交群相关的那些。在同一封信中，他还猜想了关于李代数的其他定理。Hawkins写道[3]：-

不难想象Engel读到基灵那封带有大胆猜想的信时有多么惊讶。这里是一位远在东普鲁士、致力于培养神职人员的中学里的默默无闻的教授，却在权威地论述并猜想关于索菲斯·李的变换群理论的深刻定理，而这个理论似乎是一个只有相对少数数学家知道、被更少人掌握的数学领域。

基灵于1886年夏天在前往海德堡的途中，在莱比锡拜访了Engel和索菲斯·李。此时基灵是布劳恩斯贝格Lyceum Hosianum的校长，以此身份他正在访问海德堡的姊妹机构。他于7月31日抵达莱比锡，索菲斯·李是那里的教授，Engel是dozent。这次访问并不是特别富有成果，因为尽管三人本应有丰富的数学思想可供讨论，但基灵和索菲斯·李之间似乎存在性格冲突。在莱比锡期间，基灵还会见了伊赛·舒尔和爱德华·斯图迪。接着前往海德堡度过八月，基灵那一年几乎没有做进一步的数学工作，因为他在回到布劳恩斯贝格后开始担心一个女儿的健康。

基灵于1887年4月27日写信给Engel，他已经提出了半单李代数的定义（他关于这样的代数没有阿贝尔理想的定义等价于这样的代数没有可解理想的定义）。到他在23日写信给Engel时，May Killing已经发现他关于单代数的猜想是错误的，因为他发现了$G$，而到10月18日，他已经发现了单代数的完整列表。然而，他没有这些代数的具体表示。这些结果的发表在基灵的论文Die Zusammensetzung der stetigen/endlichen TransformationsgruppenⓉ（连续/有限变换群的构成）的第三和第四部分中，上文已提及。这项工作最引人注目的部分是他对例外单李代数的发现。Helgason写道[5]：-

例外单李代数是基灵论文最后第18节的主题。这无疑是他最非凡的发现，尽管这些代数最初在他看来是一种麻烦，他努力试图消除它们。……它们后来在李理论中扮演了重要角色……

最后，在我们结束对基灵工作的讨论之前，值得注意的是他引入了矩阵的“特征方程”这一术语。

正是埃利·嘉当，在他1894年提交的学位论文中，找到了所有例外单李代数的具体表示（尽管他在论文中没有详细展开所有细节）。他还重新整理了基灵的证明，使其更容易理解。在许多方面，埃利·嘉当如此成功地在严谨而完整的单一著作中呈现了基灵对半单李代数的分类，以至于基灵因其非凡成就而没有得到人们可能预期的那么多赞誉。

我们回到对基灵职业生涯最后阶段的描述。1892年，他回到明斯特担任数学教授，并在那里度过了余生，埋头于教学、行政和慈善工作。1897至1898年，他担任明斯特大学校长。他一贯维护传统，厌恶变革。一个例子是他希望哲学对所有研究生保持必修。他竭力争取保留哲学考试，尽管正如Engel所说：-

基灵看不出对大多数考生来说，哲学考试完全毫无价值。

1900年，基灵被喀山物理数学会授予罗巴切夫斯基奖，以资表彰。这是该奖的第二次颁发，第一次在1897年授予了索菲斯·李。

1918年后德国社会凝聚力的崩溃，在基灵最后的岁月里给他带来了巨大痛苦，因为他是一个伟大的爱国者。他已经经历了两个幼子夭折的损失，但更具毁灭性的是他剩余两个儿子的失去，其中一个在1910年去世，当时正在为他的教授资格论文（Habilitation）研究一个音乐史主题，另一个在军营中生病，于1918年第一次世界大战结束前不久去世。

Coleman在[1]中写道：-

基灵一生都表现出高度的责任感，对任何在身体或精神上有需要的人都深切关怀。他深受数学家Engel所描述的“1850年代和1860年代严谨的威斯特伐利亚天主教”的熏陶。亚西西的圣方济各是他的榜样，因此他在39岁时与妻子一起加入了方济各第三会。他的学生热爱并钦佩基灵，因为他毫不吝惜地把时间和精力奉献给他们，从不满足于让他们成为狭隘的专家，因此他的讲座涵盖了几何和群之外的许多主题。

这让基灵看起来几乎是一位数学圣人，但这可能太过分了。他确实缺乏幽默感，而且他[3]：-

……对批评极其敏感。

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.