Sophus Lie chased a grand dream for equations and built modern physics' toolkit
Sophus Lie wanted to do for differential equations what Galois had done for polynomial ones: sort them all by their symmetries. That dream never fully came true. But the continuous groups he invented along the way, now called Lie groups, became the language physicists use for symmetry, from spinning particles to the quark model.
A Lie group joins two ideas. A group is a set of transformations that can be combined and undone, and a manifold is a space that looks flat when you zoom in close enough. A Lie group is both at once, with combining and undoing its elements smooth enough for calculus. The simplest picture is a circle's rotations: turn it by any angle and it looks the same, and those turns, chained together, form the circle group.
Lie, a Norwegian who lived from 1842 to 1899, dated the birth of his theory to the winter of 1873 to 1874, after years of intense work that included near-daily meetings with Felix Klein from 1869 through 1872 in Berlin, Paris and elsewhere. For much of the 1870s he published in Norwegian journals, which slowed recognition in the rest of Europe. In 1884 the young German Friedrich Engel joined him to write a systematic account, which appeared in three volumes between 1888 and 1893, and the French phrase groupes de Lie was first used in 1893 by his student Arthur Tresse.
Others turned the idea into a structure theory. Wilhelm Killing began a series of papers in 1888 that, refined by Elie Cartan, led to the classification of the semisimple kind. Hermann Weyl described their representations, separated Lie's infinitesimal groups, today's Lie algebras, from the groups themselves, and tied the subject to quantum mechanics. David Hilbert set a challenge for the field as his Fifth Problem in 1900.
Today these groups sit at the core of geometry and physics. Klein's Erlangen program defined each geometry by the transformations it leaves unchanged. In particle physics, the rotation group, the special unitary group SU(3) and the Poincare group of spacetime symmetries are among the most important, and algebraic versions developed in the 1940s and 1950s reach into number theory.
Source: Lie group
