Sophus Lie and the Role of Lie Groups in Mathematics

As Lie group theory has developed it has also become more and more pervasive in its influence on other mathematical disciplines. The original founder of this theory was a Norwegian, Marius Sophus Lie, who was born in Nordfjordeid, 1842. In order to understand the background and motivation to Lie's work we must go further back. A central problem in algebra at the end of the 18th century was that of solving algebraic equations. While 2nd, 3rd and 4 th degree equations could be solved explic­ itly by radicals it was suspected, particularly through the work of Lagrange (1771), that the general 5th degree equation could not be solved in this way. Ruffini, in 1813, proposed a proof of this; however the proof was generally found to be unsat­ isfactory. Abel gave another proof in 1824 which after subsequent repairs has been considered complete. But to Galois belongs the far-reaching idea (around 1830) of attaching to the equation a certain finite permutation group (of the roots), now called the Galois group. A remarkable theorem in Galois theory states that the solvability of this group is equivalent to the solvability of the equation by radicals. The equation x 5 x= 0 has Galois group S5, the symmetric group of five letters which is not solvable; thus the Ruffini-Abel result follows. When Sylow gave a lecture on these matters at the University of Oslo 1863, a farmer's son, by the name of Sophus Lie (1842-1899), was in the audience. Al­ though his interests were oriented more towards Geometry than Algebra, Galois' ideas made a great impression on him. After his friendly and productive collab­ oration with Klein 1970-71, Lie conceived the idea of developing an analog for differential equations to Galois theory for algebraic equations. I shall try to explain the foundations of this theory. The differential equation