ABSTRACT
This chapter describes the basic facts of Lie’s theory of continuous groups in a form that is suited for the applications to differential equations in later parts of this book. Originally Lie considered this only as an auxiliary subject for his main goal, i.e., solving differential equations in closed form by analogy with Galois’ theory of algebraic equations. After recognizing the fundamental importance of these groups for this latter problem he developed a fairly complete theory for them. Its original objective, solving differential equations, was almost completely forgotten, most textbooks on differential equations do not even mention it at all. Later on the theory of continuous groups became a field of independent interest under the name Lie groups. The same is true for the algebraic objects introduced by Killing that were baptized Lie algebras by Hermann Weyl. They were obtained by abstraction from the commutators of vector fields occurring in Lie’s theory.
