ThatsMaths

“It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002).

Sophus Lie (1842-1899)

Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

Lie’s mathematical career did not take off until, aged about 26, he discovered works on modern geometry by Plücker and Poncelet. Following his early research, Lie was awarded a travel scholarship. In Berlin he met another young geometer, Felix Klein, and began a long and fruitful collaboration with him. They were both in Paris when the Franco-Prussian war broke out. Klein returned to Berlin and Lie decided to walk southward to visit another mathematician in Milan. He was arrested en route: his mathematical papers and his eccentric behaviour were enough to arouse suspicions that he was a German spy. He was imprisoned for a month until a French colleague, Gaston Darboux, vouched for his innocence.

Back to Norway

Lie returned to Norway where he began the study of the transformation groups that now bear his name. Klein sent one of his students, Friedrich Engel (not Engels!), to study with Lie and thus began another fruitful collaboration and friendship.

The idea of a group emerged from the work of Evariste Galois on the solutions of polynomial equations. Lie wanted to do for differential equations what Galois had done for algebraic equations. This was a move from the discrete to the continuous. Polynomial equations such as quadratics and cubics have a finite number of solutions, and the corresponding groups introduced by Galois are finite. Differential equations have a continuous spectrum of solutions and the groups that Lie devised to describe these solutions are also continuous.

Lie and Klein

Later (1886) Lie took up the professorship in Leipzig that Klein had vacated upon moving to Göttingen. There he completed, with the help of Engel, a three volume treatise, The Theory of Transformation Groups. But life in Leipzig was hard and Lie suffered from homesickness. The stressful conditions may have contributed to his nervous breakdown in 1889. Although he recovered within a year, he became embittered, irascible and paranoid. Sadly, this led to a major conflict over priority with Klein. Lie returned to Oslo but his health deteriorated and within a year he was dead, aged only fifty-six.

One of Lie’s great achievements was the discovery that his continuous transformation groups could be understood by linearising them to produce what are now called Lie algebras. The German mathematician Hermann Weyl showed how Lie’s work plays a role in quantum mechanics. Today, Lie theory is central in quantum field theory. The group concept models the structure of sub-atomic particles and has enabled the prediction of completely new particles.

Preview: A Lie group is really two distinct but interconnected things, a group and a manifold. But more about that in a future post.

Sources

James, Ioan, 2002. Remarkable Mathematicians from Euler to von Neumann. Cambridge Univ. Press., 433pp.

Stillwell, John, 2008: Naive Lie Theory. Springer, 234pp.