26 2. Basic notions of representation theory
2.10. Historical interlude: Sophus Lie’s trials
and transformations
To call Sophus Lie (1842–1899) an overachiever would be an under-
statement. Scoring first at the 1859 entrance examinations to the
University of Christiania (now Oslo) in Norway, he was determined
to finish first as well. When problems with his biology class derailed
this project, Lie received only the second-highest graduation score.
He became depressed, suffered from insomnia, and even contemplated
suicide. At that time, he had no desire to become a mathematician.
He began working as a mathematics tutor to support himself, read
more and more on the subject, and eventually began publishing re-
search papers. He was 26 when he finally decided to devote himself
to mathematics.
The Norwegian government realized that the best way to edu-
cate their promising scientists was for them to leave Norway, and
Lie received a fellowship to travel to Europe. Lie went straight to
Berlin, a leading European center of mathematical research, but the
mathematics practiced by local stars Weierstrass and Kronecker
did not impress him. There Lie met young Felix Klein, who eagerly
shared this sentiment. The two had a common interest in line geome-
try and became friends. Klein’s and Lie’s personalities complemented
each other very well. As the mathematician Hans Freudenthal put it,
“Lie and Klein had quite different characters as humans and math-
ematicians: 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 generalization”
[16, p. 323].
Lie liked to bounce ideas off his friend’s head, and Klein’s returns
were often quite powerful. In particular, Klein pointed out an anal-
ogy between Lie’s research on the tetrahedral complex and the Galois
theory of commutative permutation groups. Blissfully unaware of
the difficulties on his path, Lie enthusiastically embraced this sugges-
tion. Developing a continuous analog of the Galois theory of algebraic
equations became Lie’s id´ ee fixe for the next several years.
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