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.