Theory of Lie Groups

IN recent years great advances have been made in our knowledge of the fundamental structures of analysis, particularly of algebra and topology, and an exposition of Lie groups from the modern point of view is timely. This is given admirably in the book under review, which could well have been called “Lie Groups in the Large”. To make such a treatment intelligible it is necessary to re-examine from the new point of view such subjects as the classical linear groups, topological groups and their underlying topological spaces, and analytic manifolds. Excellent accounts of these, with the study of integral manifolds and their constructions in the large, are given in the earlier chapters. In Chapter 4 we come to the definition of an analytic group and its underlying manifold, and to the introduction of the important concept of a Lie algebra associated with an analytic group. A Lie group is then denned as a locally connected topological group related in a certain way to some analytic group; from this it follows that the concept of Lie algebra can be used freely when dealing with Lie groups. Chapter 5 contains an account of Cartan's calculus of exterior differential forms and its application to the theory of Lie groups. The final chapter is concerned with compact Lie groups, and after a brief consideration of the general theory of representations it is proved, among other things, that every representation of a compact Lie group is semi-simple.Theory of Lie Groups IBy Claude Chevalley. (Princeton Mathematical Series.) Pp. xii + 217. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1946.) 20s. net.