28 2. Basic notions of representation theory

advances with his id´ ee fixe, Lie finally acquired the mathematical

weaponry needed to answer the challenge of the Erlangen Program

and to tackle the theory of continuous transformation groups.

Living on the outskirts of Europe, Lie felt quite marginalized in

the European mathematics community. No students and very few for-

eign colleagues were interested in his research. He wrote his papers

in German but published them almost exclusively in Norwegian jour-

nals, preferring publication speed over wide accessibility. A few years

later he learned, however, that one French mathematician had won

the Grand Prix from the Acad´ emie des Sciences for independently

obtained results that yielded some special cases of Lie’s work on dif-

ferential equations. Lie realized that his Norwegian publications were

not the greatest publicity vehicle, and that he needed to make his

work better known in Europe. “If only I could collect together and

edit all my results,” he wistfully wrote to Klein [22, p. 77]. Klein’s

practical mind quickly found a solution. Klein, who then taught at

Leipzig, arranged for the young mathematician Friedrich Engel, a re-

cent doctoral student of his colleague, to go to Christiania and to

render Lie a helping mathematical hand.

Lie and Engel met twice daily for a polite conversation about

transformation groups. As Engel recalled, Lie carried his theory al-

most entirely in his head and dictated to Engel an outline of each

chapter, “a sort of skeleton, to be clothed by me with flesh and blood”

[22, p. 77]. Lie read and revised Engel’s notes, eventually producing

the first draft of a book-length manuscript.

When Klein left Leipzig to take up a professorship at G¨ottingen,

he arranged for the vacated chair of geometry to be offered to Lie.

Lie somewhat reluctantly left his homeland and arrived at Leipzig

with the intention of building “a healthy mathematical school” there

[22, p. 226]. He continued his collaboration with Engel, which cul-

minated in the publication of their joint three-volume work, Theorie

der Transformationsgruppen.

Lie’s ideas began to spread around Europe, finding a particularly

fertile ground in Paris. Inspired by Lie, Henri Poincar´ e remarked that

all mathematics was a tale about groups, and

´

Emile Picard wrote to

Lie, “Paris is becoming a center for groups; it is all fermenting in