Invariant Differential Equations on Homogeneous Manifolds by Sigurdur Helgason

1. Historical origins of Lie group theory. Nowadays when Lie groups enter in a profound way into so many areas of mathematics, their historical origin is of considerable general interest. The connection between Lie groups and differential equations is not very pronounced in the modern theory of Lie groups, so in this introduction we attempt to describe some of the foundational work of S. Lie, W. Killing and E. Cartan at the time when the interplay with differential equations was significant. In fact, the actual construction of the exceptional simple Lie groups seems to have been accomplished first by means of differential equations. Although motion groups in R3 had occurred in the work of C. Jordan prior to 1870, Lie group theory as a general structure theory for the transformation groups themselves originated around 1873 with Lie's efforts about that time to use group theoretic methods on differential equations as suggested by Galois' theory for algebraic equations. It seems that a lecture by Sylow in 1863 (when Lie was 20) on Galois theory2 (Lie and Engel [9, vol. 3, p. XXII]) and his collaboration with F. Klein, 1870, on curves and transformations (Klein and Lie [6], Engel [3b, p. 35]) were particularly instrumental in suggesting to him the following: