**Sophus Lie** was a Norwegian mathematician who made major advances in the theory of continuous groups of transformations and differential equations. Lie groups and Lie algebras are named after him.

- 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.
- 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.
- 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.
- It was during the year 1867 that Lie had his first brilliant new mathematical idea.
- 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.
- 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.
- 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.
- 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.
- 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.
- In the spring of 1870 Lie and Klein were together again in Paris.
- 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.
- While in Paris Lie discovered contact transformations.
- However, Lie was a Norwegian and he was finding mathematical discussions in Paris very stimulating.
- 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.
- Only after the intervention of Darboux was Lie released from prison.
- 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.
- 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.
- Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations.
- 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.
- Klein, realising the problems, had the excellent idea of sending Friedrich Engel to Christiania to help Lie.
- 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.
- Towards the end of the 1880s Lie's relationship with Engel broke down.
- The position is complicated by the mental difficulties which Lie suffered in 1889.
- "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.
- 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.
- 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.
- 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.
- Furthermore, this evidence contradicts the oft-stated opinion that Lie's sickness was brought about by overwork.
- 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.
- 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.

Born 17 December 1842, Nordfjordeide, Norway. Died 18 February 1899, Kristiania (now Oslo), Norway.

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Prize Abel, Bourbaki, Group Theory, Origin Norway

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