Introduction

Abstract

At the end of the last century S. Lie developed the theory of “continuous transformation groups”, which gave rise to the field nowadays known as the theory of Lie groups. The work of S. Lie was to a large extent inspired by the idea of constructing the analogue of Galois theory for differential equations, but further development of the theory made clear its close relationship with other areas of mathematics (particularly with geometry) and also with theoretical physics. The authors of the present work do not attempt to give a survey of all the main results of the theory of Lie transformation groups obtained in over a century of its development. In particular almost entirely beyond the scope of this survey remain the geometry and topology of Lie groups and homogeneous spaces and the, closely connected with topology, theory of continuous actions of compact Lie groups. Special attention was paid to the general theory and to transitive actions of Lie groups, in particular, to results on the classification of transitive actions and the structure of homogeneous spaces.