Theory of Lie Groups

by Claude Chevalley

Overview

"Chevalley's most important contribution to mathematics is certainly his work on group theory. . . . [Theory of Lie Groups] was the first systematic exposition of the foundations of Lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold. This book remained the basic reference on Lie groups for at least two decades." -- Bulletin of the American Mathematical Society
Suitable for advanced undergraduate and graduate students of mathematics, this enduringly relevant text introduces the main basic principles that govern the theory of Lie groups. The treatment opens with an overview of the classical linear groups and of topological groups, focusing on the theory of covering spaces and groups, which is developed independently from the theory of paths.
Succeeding chapters contain an examination of the theory of analytic manifolds as well as a combination of the notions of topological group and manifold that defines analytic and Lie groups. An exposition of the differential calculus of Cartan follows and concludes with an exploration of compact Lie groups and their representations.

Product Details

ISBN-13:	9780486824536
Publisher:	Dover Publications
Publication date:	03/21/2018
Series:	Dover Books on Mathematics
Edition description:	Unabridged
Pages:	224
Product dimensions:	6.00(w) x 9.00(h) x (d)

About the Author

Claude Chevalley (1909-1984) served on the faculty of Princeton University and was resident at the Institute for Advanced Study. He was a member of the Bourbaki and was awarded the Cole Prize of the American Mathematical Society.

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INTRODUCTION

Expository books on the theory of Lie groups generally confine themselves to the local aspect of the theory. This limitation was probably necessary as long as general topology was not yet sufficiently well elaborated to provide a solid base for a theory in the large. These days are now passed, and we have thought that it would be useful to have a systematic treatment of the theory from a global point of view. The present volume introduces the main basic principles which govern the theory of Lie groups.

A Lie group is at the same time a group, a topological space and a manifold: it has therefore three kinds of "structures," which are interrelated with each other. The elementary properties of abstract groups are by now sufficiently well known to the general mathematical public to make it unnecessary for such a book as this one to contain a purely group-theoretic chapter. The theory of topological groups, however, has been included and is treated in Chapter II. The greatest part of this chapter is concerned with the theory of covering spaces and groups, which is developed independently from the theory of paths. Chapter III is concerned with the theory of (analytic) manifolds (independently of the notion of group). Our definition of a manifold is inspired by the definition of a Riemann surface given by H. Weyl in his book "Die Idee der Riemannschen Flache"; it has, compared with the definition by overlapping system of coordinates, the advantage of being intrinsic. The theory of involutive systems of differential equations on a manifold is treated not only from the local point of view but also in the large. In order to achieve this, a definition of the submanifolds of a manifold is given according to which a submanifold is not necessarily a topological subspace of the manifold in which it is imbedded.

The notions of topological group and manifold are combined together in Chapter IV to give the notions of analytic group and Lie group. An analytic group is a topological group which is given a priori as a manifold; a Lie group (at least when it is connected) is a topological group which can be endowed with a structure of manifold in such a way that it becomes an analytic group. It is shown that, if this is possible, the manifold-structure in question is uniquely determined, so that connected Lie groups and analytic groups are in reality the same things defined in different ways. We shall see however in the second volume that the difference becomes a real one when complex analytic groups are considered instead of the real ones which are treated here.

Chapter V contains an exposition of the theory of exterior differential forms of Cartan which plays an essential role in the general theory of Lie groups, as well in its topological as in its differential geometric aspects. This theory leads in particular to the construction of the invariant integral on a Lie group. In spite of the fact that this invariant integration can be defined on arbitrary locally compact groups, we have thought that it is more in the spirit of a treatise on Lie groups to derive it from the existence of left invariant differential forms.

Chapter VI is concerned with the general properties of compact Lie groups. The fundamental fact is of course contained in the statement of Peter-Weyhs theorem which guarantees the existence of faithful linear representations. We have also included a proof of the generalization by Tannaka of the Pontrjagin duality theorem. A slight modification of the original proof of Tannaka shows that a compact Lie group may be considered as the set of real points of an algebraic variety in a complex affine space, the whole variety being itself a Lie group on which complex coordinates can be introduced.

The second volume of this book, now in preparation, will be mainly concerned with the theory and classification of semi-simple Lie Groups.

In preparing this book, I have received many valuable suggestions from several of my friends, in particular from Warren Ambrose, Gerhardt Hochschild, Deane Montgomery and Hsiao Fu Tuan. I was helped in reading the proofs by John Coleman and Norman Hamilton. I have also received precious advice from Professor H. Weyl and Professor S. Lefschetz. To all of them I am glad to express here my deep gratitude.

C. C.

(Continues…)

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Excerpted from "Theory of Lie Groups"
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Copyright © 2018 Claude Chevalley.
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Table of Contents

Table of Contents:
Introduction
I. The Classical Linear Groups
II. Topological Groups
III. Manifolds
IV. Analytic Groups. Lie Groups
V. The Differential Calculus of Cartan
VI. Compact Lie Groups and Their Representations
Index.

Editorial Reviews

"Chevalley's most important contribution to mathematics is certainly his work on group theory . . . [Theory of Lie Groups] was the first systematic exposition of the foundations of Lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold. This book remained the basic reference on Lie groups for at least two decades."

Bulletin of the American Mathematical Society