索菲斯·李Sophus Lie

索菲斯·李的父亲是Johann Herman Lie，一位路德宗牧师。他的父母有六个孩子，索菲斯·李是六个中最小的。索菲斯·李最初在莫斯镇上学，这是挪威东南部的一个港口，位于奥斯陆峡湾东侧。1857年，他进入克里斯蒂安尼亚的尼森私立拉丁学校（这座城市后来成为克里斯蒂安尼亚，1925年又成为奥斯陆）。在这所学校期间，他决定投身军旅，但视力不够好，于是放弃了这个想法，进入了克里斯蒂安尼亚大学。

在大学里，索菲斯·李学习了一门广泛的科学课程。这门课程中肯定有一些数学，索菲斯·李在1862年参加了彼得·卢德维格·梅德尔·西罗的讲座。虽然不是永久教职人员，彼得·卢德维格·梅德尔·西罗教授了一门课程，替代Broch，他在其中解释了尼尔斯·阿贝尔和埃瓦里斯特·伽罗瓦关于代数方程的工作。索菲斯·李还参加了卡尔·安东·皮耶克尼斯关于数学的讲座，所以他肯定有相当质量的老师，然而他在1865年毕业时，没有表现出对该学科的任何巨大能力，也没有任何特别的喜爱。

随后有一段时期，索菲斯·李无法决定追求什么学科，他在尝试做出决定的同时教学生。他知道自己想要的一件事是学术生涯，他一度认为天文学可能是正确的主题。他学了一些力学，想知道植物学、动物学或物理学是否可能是正确的学科，总的来说变得相当困惑。然而，有迹象表明，从1866年起，他开始阅读越来越多的数学，克里斯蒂安尼亚大学的图书馆记录清楚地显示，他的兴趣正稳步转向那个方向。

正是在1867年，索菲斯·李有了他第一个辉煌的新数学想法。他在半夜想到了这个想法，兴奋不已，冲去见他的朋友Ernst Motzfeldt，叫醒他并喊道：-

我找到了，它很简单！

当然，这并非索菲斯·李问题的终结（远非如此，因为索菲斯·李总会有问题），但至少在他自己心中，他现在知道自己想要从事的职业，可以说从那一刻起，索菲斯·李成为了一名数学家。索菲斯·李所研究的数学类型在1868年变得更加明确，当时他如饥似渴地阅读尤里乌斯·普吕克和让-维克托·彭赛列关于几何的论文。尤里乌斯·普吕克的[1]：-

……通过选择除点以外的图形——实际上是直线——作为空间的元素来创建新几何学的这一不朽思想贯穿了索菲斯·李的全部工作。

索菲斯·李在1869年写了一篇简短的数学论文，自费出版，其灵感来自1867年触动他的想法。他写了一份更详细的阐述，但数学界过于谨慎，未能迅速接受索菲斯·李的革命性观念。克里斯蒂安尼亚的科学院不愿发表他的作品，在这个阶段，索菲斯·李开始绝望，认为自己无法在数学界获得认可。他的朋友Motzfeldt出色地鼓励索菲斯·李继续推进他的数学思想，突破出现在1869年晚些时候，当时Crelle's Journal接受了他的论文。他写信给两位普鲁士数学家，Reye和阿尔弗雷德·克莱布什，仍然试图为自己的想法获得认可。然而，Crelle's Journal上的那篇论文被证明至关重要，因为凭借这篇论文，索菲斯·李获得了一笔奖学金，得以旅行并会见顶尖数学家。

1869年底，索菲斯·李出发前往普鲁士，访问了哥廷根，然后去了柏林。在柏林，他会见了利奥波德·克罗内克、恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯。索菲斯·李对主导柏林的卡尔·魏尔斯特拉斯的数学风格不感兴趣。他的兴趣更接近恩斯特·爱德华·库默尔，索菲斯·李在恩斯特·爱德华·库默尔的讨论班上讲授了自己的成果，并能够纠正恩斯特·爱德华·库默尔在3次线汇工作中犯的一些错误。然而，对索菲斯·李最重要的是，他在柏林遇到了菲利克斯·克莱因。很容易看出，这两人会立即在数学上找到共同点，因为菲利克斯·克莱因曾是尤里乌斯·普吕克的学生，而索菲斯·李，尽管从未见过尤里乌斯·普吕克，总是说他觉得自己像尤里乌斯·普吕克的学生。尽管通过尤里乌斯·普吕克的线几何学有共同联系，索菲斯·李和菲利克斯·克莱因在性格上相当不同，正如汉斯·弗洛伊登萨在[1]中指出的：-

索菲斯·李和菲利克斯·克莱因作为人和数学家有着相当不同的性格：代数学家菲利克斯·克莱因着迷于迷人问题的奇特之处；分析学家索菲斯·李则从特殊情况出发，寻求在适当的推广中理解问题。

正是在柏林，索菲斯·李对自己的数学能力产生了新的自信。他得到了恩斯特·爱德华·库默尔的高度赞扬，并收到了Reye和阿尔弗雷德·克莱布什对他早先信件的回复，这极大地鼓舞了他。索菲斯·李写信给他在克里斯蒂安尼亚的朋友Motzfeldt说（例如见[34]）：-

在1864-68年间，我确实低估了自己的智力。

1870年春天，索菲斯·李和菲利克斯·克莱因再次相聚于巴黎。在那里他们遇到了让·加斯东·达布、米歇尔·沙勒和卡米耶·若尔当。卡米耶·若尔当似乎以彼得·卢德维格·梅德尔·西罗未能做到的方式取得了成功，因为卡米耶·若尔当使索菲斯·李意识到group theory对几何学研究的重要性。索菲斯·李开始发展后来出现在他关于变换群的工作中的思想。他开始与菲利克斯·克莱因讨论这些关于群和几何的新思想，后来与菲利克斯·克莱因合作发表了几篇论文。这项合作工作的成果之一是菲利克斯·克莱因在1872年的Erlangen Program中将几何刻画为在群作用下不变的性质。而在Paris Lie发现了接触变换。这些变换使得直线与球面之间可以建立一一对应，使得切线球面对应于相交直线。

索菲斯·李 和 菲利克斯·克莱因 在巴黎深入思考数学，而法国与普鲁士之间的政治局势正在恶化。法国皇帝拿破仑三世在法国的声望正在下降，他认为与普鲁士的战争可能会改变他的政治命运，因为他的顾问告诉他法国军队能够击败普鲁士。普鲁士首相俾斯麦则将与法国的战争视为统一南德意志各邦的机会。由于双方都认为战争对自己有利，普法战争变得不可避免。7月14日，俾斯麦发出了一封电报，激怒了法国政府，7月19日法国对普鲁士宣战。对于 菲利克斯·克莱因 来说，一个在宣战时恰好在巴黎的普鲁士公民，只有一种可能：他必须迅速返回柏林。

索菲斯·李 是挪威人，他发现巴黎的数学讨论非常令人振奋。他决定留下，但随着德军进攻只遇到法国无效的回应，他变得焦虑起来。8月，德军将部分法国军队困在梅斯，索菲斯·李 决定是时候离开了，他计划徒步前往意大利。他到达了枫丹白露，但在那里被当作德国间谍逮捕，他的数学笔记被假定为绝密编码信息。只有在 让·加斯东·达布 的干预下，索菲斯·李 才从监狱获释。法国军队已于9月1日投降，9月19日德军开始封锁巴黎。索菲斯·李 再次逃往意大利，然后从那里经德国返回克里斯蒂安尼亚，以便与 菲利克斯·克莱因 会面并讨论数学。

1871年，索菲斯·李 获得奖学金后成为克里斯蒂安尼亚的助理，他还在克里斯蒂安尼亚的尼森私立拉丁学校任教，那里曾是他自己就读的学校。他提交了一篇学位论文 On a class of geometric transformations（用挪威语写成）以获得博士学位，该学位于1872年7月正式授予。这篇学位论文包含了他发表在 Crelle's Journal 上的最初成果中的思想，以及关于接触变换的工作，这些变换的一个特例是将线映射为球的变换，这是他在巴黎时发现的。

显然，索菲斯·李 是一位杰出的数学家，克里斯蒂安尼亚大学以非常积极的方式做出回应，于1872年为他设立了一个讲席。著名的挪威数学家 尼尔斯·阿贝尔 在此前40多年（大约在 索菲斯·李 出生前14年）就去世了，但尽管 尼尔斯·阿贝尔 的职业生涯短暂，他的全集当时尚未出版。挪威数学家承担这项任务是自然的，在1873年至1881年间，彼得·卢德维格·梅德尔·西罗 和 索菲斯·李 准备了 尼尔斯·阿贝尔 全集的版本。然而，索菲斯·李 总是声称大部分工作是由 彼得·卢德维格·梅德尔·西罗 完成的。在 索菲斯·李 被任命为讲席后两年内发生的另一件事是他的婚姻。他与 Anna Birch 结婚，他们会有三个孩子，一个女儿和两个儿子。

索菲斯·李已开始研究偏微分方程，希望找到一种类似于方程伽罗瓦理论的理论。他写道：-

……微分方程理论是现代数学中最重要的学科。

他研究了自己的接触变换，考虑它们如何影响由卡尔·古斯塔夫·雅各布·雅可比提出的从给定微分方程生成更多解的过程。这导致以索菲斯·李称为infinitesimal group的方式组合这些变换，但按我们的定义这不是一个群，而是今天所称的李代数。正是在1873-74年冬季，索菲斯·李开始系统地发展后来成为其连续变换群理论的内容，后来被称为李群，放弃了他最初研究偏微分方程的意图。后来威廉·基灵研究了与索菲斯·李群相关的索菲斯·李代数。他相当独立于索菲斯·李完成了这项工作（而且看来并非以索菲斯·李觉得满意的方式），正是埃利·嘉当在1900年完成了半单索菲斯·李代数的分类。

索菲斯·李正在创造极具创新性的数学，但他越来越悲伤，因为他在数学界得不到认可。一个原因无疑是他在克里斯蒂安尼亚的孤立，但第二个原因是他的论文不容易理解，部分是由于他的写作风格，部分是因为他的几何直觉远远超过其他数学家。菲利克斯·克莱因意识到这些问题，有一个绝妙的主意，派Friedrich Engel去克里斯蒂安尼亚帮助索菲斯·李。

Engel于1883年在莱比锡获得博士学位，师从阿道夫·迈尔，写了一篇关于接触变换的学位论文。菲利克斯·克莱因认识到他是协助索菲斯·李的合适人选，并在菲利克斯·克莱因的建议下，Engel于1884年开始在克里斯蒂安尼亚与索菲斯·李一起工作。他与索菲斯·李一起工作了九个月，于1885年离开。Engel随后被任命到莱比锡，当菲利克斯·克莱因于1886年离开莱比锡的讲席时，索菲斯·李被任命接替他。Engel和索菲斯·李之间的合作持续了九年，最终在1888年至1893年间出版了三卷本的联合重要著作Theorie der Transformationsgruppen。这是索菲斯·李关于连续变换群的主要著作。

在莱比锡，索菲斯·李的生活与在克里斯蒂安尼亚时颇为不同。他现在处于数学主流之中，学生从许多国家前来跟随他学习。然而，他的教学负担要重得多[38]：-

索菲斯·李关于自己研究的讲座受到学生的高度评价，相比之下，他那些关于标准主题的必修讲座则不太受欢迎。……他宁愿画一张图，也不愿给出严格的证明。

然而一切并不顺利，他仍然感到未受认可，正如Svare在[38]中所写：-

索菲斯·李为持续的思乡之情所困扰。作为一个热爱户外的人，他怀念挪威的森林和山脉。

到了19世纪80年代末，索菲斯·李与Engel的关系破裂了。1892年，索菲斯·李与菲利克斯·克莱因之间终生的友谊破裂，次年索菲斯·李公开攻击菲利克斯·克莱因，说：-

我不是菲利克斯·克莱因的学生，反过来也不是，尽管后者可能更接近事实。

任何传记作者都很难公正地描述这些事件以及随后发生的事件，因为文献中存在大量相互矛盾的材料。其原因不难理解，因为多年来关于索菲斯·李的信息都基于Engel在索菲斯·李去世时写的[13]。索菲斯·李在1889年遭受的精神困扰使情况更加复杂。菲利克斯·克莱因的[34]：-

……通过援引天才与疯狂之间的密切关系来为索菲斯·李的行为进行“辩护”，确实制造出一种至今仍被普遍接受的解释。通过这一“辩护”行为，菲利克斯·克莱因对他的老朋友造成了令人难以置信的不公。

事实是，索菲斯·李的行为并不像人们所描绘的那样完全失去理性，而确实是受到Engel和菲利克斯·克莱因两人行为方式的驱使。Purkert在[26]中讨论了索菲斯·李与Engel之间关系的破裂。他研究了来自莱比锡大学的材料，并认为索菲斯·李改变了对Engel的态度，是因为索菲斯·李仍然感到缺乏认可，但他知道自己作为富有创造力的数学家与Engel不在同一层次。索菲斯·李于1898年回到克里斯蒂安尼亚，担任专为他设立的职位。他就谁应填补他的讲席写了一份报告，该报告全文见[26]。尽管Engel是索菲斯·李自己研究领域中的主要工作者之一，Purkert认为索菲斯·李关于他缺乏创造力的评价完全公正。

在[15]中，Fritzsche评论了索菲斯·李的疾病。他写道：-

通过关于索菲斯·李疾病的信息，可以追溯出一些后果，这些后果揭示了他生平中某些传记方面的内容；例如，他与Friedrich Engel和菲利克斯·克莱因的决裂。此外，这一证据与那种常被提及的观点相矛盾，即索菲斯·李的疾病是由过度劳累引起的。

Straume在[34]中指出，为什么索菲斯·李对菲利克斯·克莱因的行为，以及1892年的最终决裂，并非失去理性：-

Klein 1872年的“埃尔朗根纲领”并未引起太多关注；事实上，影响该纲领所设想的数学发展的是索菲斯·李，而不是菲利克斯·克莱因本人。……菲利克斯·克莱因决定重新发表该纲领，并撰写其起源（索菲斯·李在其中参与甚多），但索菲斯·李强烈不同意菲利克斯·克莱因关于过去所发生之事的看法。结果还发现，菲利克斯·克莱因烧毁了他从索菲斯·李那里收到的直到1877年的所有信件（从而打破了他们之间先前的共同约定）。

索菲斯·李在1893年出版的Theorie der Transformationsgruppen第三卷前言中公开攻击菲利克斯·克莱因作为回应。当然，索菲斯·李是个愤怒的人，但他攻击的是在世界数学舞台上占据如此领导地位的人，因此这种攻击总是更可能反弹到索菲斯·李身上，而不是伤害菲利克斯·克莱因。当前的研究已经表明，在这件事上索菲斯·李比先前报道的要好得多（因此菲利克斯·克莱因则不那么好），所有迹象都表明，进一步的研究将证明对索菲斯·李更为有利。

也许索菲斯·李热爱自己祖国的一个迹象是，他从1872年首次受聘起就一直保留着在Christiania的讲席，在莱比锡持有讲席期间则正式休假。然而，当他在1898年回到Christiania的一个讲席时，他的健康已经在恶化，并于1899年2月就任该职后不久死于恶性贫血。

让我们以引用Robert Hermann为[4]所写的前言来结束：-

在为这些翻译的评论做准备而阅读索菲斯·李的著作时，我被源自索菲斯·李工作的几何思想的丰富性和美感所震撼。其中只有一小部分被吸收到主流数学中。他以宏大的措辞思考和写作，这种风格现在已经过时，并且会被我们的科学期刊审查！这里翻译的论文以及我们翻译的后续卷中呈现的索菲斯·李，是他最狂野和最伟大的形式。

Sophus Lie's father was Johann Herman Lie, a Lutheran minister. His parents had six children and Sophus was the youngest of the six. Sophus first attended school in the town of Moss, which is a port in south-eastern Norway, on the eastern side of the Oslo Fjord. In 1857 he entered Nissen's Private Latin School in Christiania (the city which became Kristiania, then Oslo in 1925) . While at this school he decided to take up a military career, but his eyesight was not sufficiently good so he gave up the idea and entered University of Christiania.

At university Lie studied a broad science course. There was certainly some mathematics in this course, and Lie attended lectures by Ludwig Sylow in 1862. Although not on the permanent staff, Sylow taught a course, substituting for Broch, in which he explained Abel's and Galois' work on algebraic equations. Lie also attended lectures by Carl Bjerknes on mathematics, so he certainly had teachers of considerable quality, yet he graduated in 1865 without having shown any great ability for the subject, or any great liking for it.

There followed a period when Lie could not decide what subject to pursue and he taught pupils while trying to make his decision. The one thing he knew he wanted was an academic career and he thought for a while that astronomy might be the right topic. He learnt some mechanics, wondered whether botany or zoology or physics might be the right subjects and in general became rather confused. However, there are signs that from 1866 he began to read more and more mathematics and the library records in the University of Christiania show clearly that his interests were steadily turning in that direction.

It was during the year 1867 that Lie had his first brilliant new mathematical idea. It came to him in the middle of the night and, filled with excitement, he rushed to see his friend Ernst Motzfeldt, woke him up and shouted:-

I have found it, it is quite simple!

This was not the end of Lie's problems of course (far from it for Lie would always have problems), but at least in his own mind he now knew the career he wanted and it would be fair to say that from that moment on Lie became a mathematician. The type of mathematics that Lie would study became more clearly defined during 1868 when he avidly read papers on geometry by Plücker and Poncelet. Plücker's [1]:-

... monumental idea to create new geometries by choosing figures other than points - in fact straight lines - as elements of space pervaded all of Lie's work.

Lie wrote a short mathematical paper in 1869, which he published at his own expense, based on the inspiration which had struck him in 1867. He wrote up a more detailed exposition, but the world of mathematics was too cautious to quickly accept Lie's revolutionary notions. The Academy of Science in Christiania was reluctant to publish his work, and at this stage Lie began to despair that he would become accepted in the mathematical world. His friend Motzfeldt did a superb job of encouraging Lie to press on with his mathematical ideas and the breakthrough came later in 1869 when Crelle's Journal accepted his paper. He sent letters to two Prussian mathematicians, Reye and Clebsch, still attempting to gain recognition for his ideas. The paper in Crelle's Journal, however, proved vital for, on the strength of the paper, Lie was awarded a scholarship to travel and meet the leading mathematicians.

Setting off near the end of the year 1869, Lie went to Prussia and visited Göttingen and then Berlin. In Berlin he met Kronecker, Kummer and Weierstrass. Lie was not attracted to the style of Weierstrass's mathematics which dominated Berlin. His interests fitted more closely with Kummer, and Lie lectured on his own results in Kummer's seminar and was able to correct some errors that Kummer had made in his work on line congruences of degree 3. Most important to Lie, however, was the fact that in Berlin he met Felix Klein. It was easy to see that these two would instantly find common ground in mathematics since Klein had been a student of Plücker, and Lie, although he never met Plücker, always said that he felt like Plücker's student. Despite the common link through Plücker's line geometry, Lie and Klein were rather different in character as Freudenthal points out in [1]:-

Lie and Klein had quite different characters as humans and mathematicians: the algebraist Klein was fascinated by the peculiarities of charming problems; the analyst Lie, parting from special cases, sought to understand a problem in its appropriate generalisation.

It was in Berlin that Lie developed a new self-confidence in his mathematical ability. He received high praise from Kummer, and he received replies from Reye and Clebsch to his earlier letters which greatly encouraged him. Lie wrote to his friend Motzfeldt in Christiania saying (see for example [34]):-

... in the years 1864-68, I really underestimated my own mental power.

In the spring of 1870 Lie and Klein were together again in Paris. There they met Darboux, Chasles and Camille Jordan. Jordan seems to have succeeded in a way that Sylow did not, for Jordan made Lie realise how important group theory was for the study of geometry. Lie started to develop ideas which would later appear in his work on transformation groups. He began to discuss with Klein these new ideas on groups and geometry and he would collaborate later with Klein in publishing several papers. This joint work had as one of its outcomes Klein's characterisation of geometry in his Erlangen Program of 1872 as properties invariant under a group action. While in Paris Lie discovered contact transformations. These transformations allowed a 1-1 correspondence between lines and spheres in such a way that tangent spheres correspond to intersecting lines.

While Lie and Klein thought deeply about mathematics in Paris, the political situation between France and Prussia was deteriorating. The popularity of Napoleon III, the French emperor, was declining in France and he thought a war with Prussia might change his political fortunes since his advisers having told him that the French Army could defeat Prussia. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states. With both sides feeling that a war was to their advantage, the Franco-Prussian War became inevitable. On 14 July, Bismarck sent a telegram which infuriated the French government and on the 19 July France declared war on Prussia. For Klein, a Prussian citizen who happened to be in Paris when war was declared, there was only one possibility: he had to return quickly to Berlin.

However, Lie was a Norwegian and he was finding mathematical discussions in Paris very stimulating. He decided to remain but became anxious as the German offensive met with only an ineffective French reply. In August, the German army trapped part of the French army in Metz and Lie decided it was time for him to leave and he planned to hike to Italy. He reached Fontainebleau but there he was arrested as a German spy, his mathematics notes being assumed to be top secret coded messages. Only after the intervention of Darboux was Lie released from prison. The French army had surrendered on 1 September, and on 19 September the German army began to blockade Paris. Lie fled again to Italy, then from there he made his way back to Christiania via Germany so that he could meet and discuss mathematics with Klein.

In 1871 Lie became an assistant at Christiania, having obtained a scholarship, and he also taught at Nissen's Private Latin School in Christiania where he had been a pupil himself. He submitted a dissertation On a class of geometric transformations (written in Norwegian) for his doctorate which was duly awarded in July 1872. The dissertation contained ideas from his first results published in Crelle's Journal and also the work on contact transformations, a special case of these transformations being a transformation which maps a line into a sphere, which he had discovered while in Paris.

It was clear that Lie was a remarkable mathematician and the University of Christiania reacted in a very positive way, creating a chair for him in 1872. The famous Norwegian mathematician Abel had died more than 40 years before this (some 14 years before Lie was born) but, despite Abel's short career, his complete works had not been published at that time. It was natural that Norwegian mathematicians would undertake the task, and between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete works. Lie, however, always claimed that most of the work was done by Sylow. Another event which took place within two years of Lie being appointed to his chair was his marriage. He married Anna Birch and they would have three children, one daughter and two sons.

Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations. He wrote:-

... the theory of differential equations is the most important discipline in modern mathematics.

He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one. This led to combining the transformations in a way that Lie called an infinitesimal group, but which is not a group with our definition, rather what is today called a Lie algebra. It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups leaving behind his original intention of examining partial differential equations. Later Killing was to examine the Lie algebras associated with Lie groups. He did this quite independently of Lie (and not it would appear in a manner which Lie found satisfactory), and it was Cartan who completed the classification of semisimple Lie algebras in 1900.

Although Lie was producing highly innovative mathematics, he became increasingly sad at the lack of recognition he was receiving in the mathematical world. One reason was undoubtedly his isolation in Christiania, but a second reason was that his papers were not easily understood, partly through his style of writing and partly because his geometrical intuition greatly exceeded that of other mathematicians. Klein, realising the problems, had the excellent idea of sending Friedrich Engel to Christiania to help Lie.

Engel had received his doctorate from Leipzig in 1883 having studied under Adolph Mayer writing a thesis on contact transformations. Klein recognised that he was the right man to assist Lie and, at Klein's suggestion, Engel went to work with Lie in Christiania starting in 1884. He worked with Lie for nine months leaving in 1885. Engel then was appointed to Leipzig and, when Klein left the chair at Leipzig in 1886, Lie was appointed to succeed him. The collaboration between Engel and Lie continued for nine years culminating with their joint major publication Theorie der Transformationsgruppen in three volumes between 1888 and 1893. This was Lie's major work on continuous groups of transformations.

In Leipzig, life for Lie was rather different from that in Christiania. He was now in the mainstream of mathematics and students came from many countries to study under him. He had a much heavier teaching load, however [38]:-

Lie's lectures on his own research were highly rated by the students, in contrast to his somewhat unpopular obligatory lectures on standard topics. ... he preferred to draw a picture instead of giving rigorous proofs.

However all was not well, he still felt unrecognised and, as Svare writes in [38]:-

In Leipzig Lie was troubled by constant homesickness. A keen outdoor man, he missed the forests and mountains of Norway.

Towards the end of the 1880s Lie's relationship with Engel broke down. In 1892 the lifelong friendship between Lie and Klein broke down and the following year Lie publicly attacked Klein saying:-

I am no pupil of Klein, nor is the opposite the case, although this might be closer to the truth.

It is difficult for any biographer to represent these events, and the events which followed, fairly since there is a great deal of contradictory material in the literature. The reason for this is not hard to understand, for information about Lie was for many years based on [13] which Engel wrote on Lie's death. The position is complicated by the mental difficulties which Lie suffered in 1889. Klein's [34]:-

... "defence" of Lie's behaviour by referring to the close relationship between genius and madness really created a generally accepted explanation which has survived up to the present. By this act of "defence" Klein did his old friend an incredible injustice.

The truth is that Lie's behaviour was not totally irrational as it has been portrayed, but was indeed motivated by the way that both Engel and Klein had behaved. Purkert in [26] discusses the breakdown of relations between Lie and Engel. He has studied material from the University of Leipzig and believes that Lie changed his attitude toward Engel because Lie still felt a lack of recognition yet he knew that he was in a different class as a creative mathematician to Engel. Lie returned to Christiania in 1898 to take up a post specially created for him. He produced a report about who should fill his chair, and this is given in full in [26]. Despite Engel being one of the leading workers in Lie's own research field, Purkert believes that Lie's assessment that he lacked creativity was entirely fair.

In [15] Fritzsche comments on Lie's illness. He writes:-

Through information about Sophus Lie's illness it is possible to trace consequences that shed light on certain biographical aspects of his life; for example, his break with Friedrich Engel and Felix Klein. Furthermore, this evidence contradicts the oft-stated opinion that Lie's sickness was brought about by overwork.

Straume in [34] points out why Lie's behaviour towards Klein, with the final breakdown in 1892, was not irrational:-

Klein's 'Erlangen Program' from 1872 had not attracted much attention; in fact, it was Lie rather than Klein himself who had influenced the mathematical development envisioned in this Program. ... Klein decided to republish the Program and also write about its origins (in which Lie was much involved), but Lie disagreed strongly with Klein's views on what had happened in the past. It also turned out that Klein burned all the letters he had received from Lie up to 1877 (and thus breaking a previous mutual agreement between them).

Lie reacted by publicly attacking Klein in the Preface to the third volume of his Theorie der Transformationsgruppen in 1893. Certainly Lie was an angry man but he was attacking someone holding such a leading role on the world scene of mathematics that the attack was always more likely to rebound on Lie rather than hurt Klein. Already current research is showing Lie in a much better light over this affair (and therefore Klein in a less good one) than previously reported and all the indications are that further research will prove even more favourable to Lie.

Perhaps an indication of Lie's love for his homeland is the fact that he continued to hold his chair in Christiania from his first appointment in 1872, being officially on leave while holding the chair in Leipzig. However his health was already deteriorating when he returned to a chair in Christiania in 1898, and he died of pernicious anaemia in February 1899 soon after taking up the post.

Let us end by quoting from Robert Hermann's preface to [4]:-

In reading Lie's work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie's work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form.