Lie Groups (I)

This brings us to the other terms of the title of this book, namely "Lie groups" and "homogeneous spaces". The theory of Lie has its roots in the study of symmetries of systems of differential equations, and the integration techniques for them. At that time, Lie had called these symmetries "continuous groups". In fact, his main goal was to develop an analogue of Galois theory for differential equations. The equations that Lie studied are now known as equations of Lie type, and an example of these is the well-known Riccati equation. Lie developed a method for solving these equations that is related to the process of "solution by quadrature". In Galois' terms, for a solution of a polynomial equation with radicals, there is a corresponding finite group. Correspondingly, to a solution of a differential equation of Lie type by quadrature, there is a corresponding continuous group.

—Andreas Arvanitoyeorgos

An Introduction to Lie Groups and

the Geometry of Homogeneous Spaces