1.1. From the Preface.

Taken in its broad sense, Lie group theory involves all of the major branches of mathematics: algebra, analysis, topology, differential geometry, and algebraic geometry. This book takes up the subject in the narrow sense, omitting differential geometry and algebraic geometry from the above list, and minimising the analysis. It is addressed primarily to the student who wants to learn the basic techniques and results of Lie group theory quickly, without having to weave his way in and out of many books and papers. The reader is assumed to have at least a superficial acquaintance with multilinear algebra, group theory, general topology, and real variable theory. This means that he should know the initial definitions and their immediate consequences, and be familiar with the grammar of mathematics in a general way. For a student at this stage, the book is technically self-contained. Its plan is to reach the most important results as directly as possible, developing the mathematical tools only to the extent to which they are used here. The development of the main theme is limited by my decision to resist the temptation to introduce and use the theory of algebraic linear groups, which would make it difficult to reach an end before the author and/or the reader is too tired to care.

Almost all of the content of this book represents standard mathematical equipment from the tool chest of the specialist. Although much of this is of comparatively recent and hence easily traceable origin, I have shied away completely from the task of revealing the original sources and giving due credit. The fact that this task is rather awkward is illustrated by the situation of Theorem 3.1 of Chapter XV. The names to be attached to this result are: É Cartan, C Chevalley, K Iwasawa, A Malcev, G D Mostow. Moreover, in the proof of its hardest part, I have made use of an unpublished simplification due to Chevalley (the proof of Theorem 1.2 of Chapter XV) of Mostow's simplification of Cartan's original proof. The only sources listed are a few of the most important books that can serve to amplify and support what is presented here and to enlarge the reader's view of the theory. One of these, Chevalley's Theory of Lie Groups, is, of course, the very foundation on which everything given here rests, although an acquaintance with it is not presupposed.

The organisation of this book is as follows. Chapter I contains the generalities concerning locally compact topological groups, and in particular the existence and uniqueness of the Haar integral. It also contains a few basic facts from general topology, such as the Stone-Weierstrass Theorem.

Chapter II contains the general theory of compact groups and their representations. The most important results here are the Peter-Weyl Theorem and the Tannaka Duality Theorem. Chapter III deals with a number of basic results on the structure of locally compact groups, chiefly abelian groups, that can be proved by elementary methods. Chapter IV develops the topological theory of covering spaces and covering groups, which plays a dominant role in Lie group theory. All the classical analysis that is used later on is given in Chapter V. Chapter VI introduces analytic manifolds. No more than the basic definitions and the main facts concerning analytic maps is given here, and no more than this is used later on.

Lie group theory proper begins with Chapter VII. The Lie algebra of an analytic group is defined, and the main properties of the exponential map of the Lie algebra into the group are established. Chapter VIII deals with the closed subgroups of analytic groups, homogeneous spaces, and factor groups. The main results are that a closed subgroup of a Lie group is a Lie group and that the factor group of an analytic group by a closed normal subgroup is an analytic group. Chapter IX establishes the technique of semidirect product constructions, which is used extensively later on. The bulk of Chapter X is the purely algebraic development of the Campbell-Hausdorff formula which is basic for the subsequent deeper analysis of the exponential map in the theory of analytic groups. From this, the theory of analytic subgroups is then readily obtained at the end of the chapter. At this stage, the road is cleared for the applications of Lie algebra theory to the theory of analytic groups.

Chapter XI is the first instalment of pure Lie algebra theory, sufficient for the first serious application, which is made in Chapter XII. The main result here is that every finite-dimensional real Lie algebra is the Lie algebra of an analytic group. The proof is based on semi-direct product constructions, and yields topological results concerning normal analytic subgroups of simply connected analytic groups, which play an important role in the structure and representation theory. Chapter XIII deals with the general structure theory of compact analytic groups. Chapter XIV brings Lie algebra theory to the stage required for the rest of this book.

1.2. Review by F Haimo.
The American Mathematical Monthly 74 (3) (1967), 342.

This excellent introduction is not written for the innocent. The exposition is at the level of the well trained advanced graduate student or young Ph.D. who wishes to inform himself quickly of a vital area. For the non-expert (e.g., the reviewer) the way is eased by the first 75 pages, the main theme appearing only in Chapter VII. Each chapter is well organised almost as an independent unit, and each is of just the right length, long enough to be informative and short enough to avoid discouraging the reader, and ends with two to six nontrivial problems. One wishes that the author would share his considerable expository talents (in, perhaps, a 350 page book) with a somewhat less knowledgeable audience.

1.3. Review by: F Hirzebruch.
Mathematical Reviews MR0207883 (34 #7696).

It is remarkable to what extent the author succeeds in providing a self-contained text for a reader familiar with the fundamentals of multilinear algebra, group theory, point-set topology, and calculus.
...

The book originated from a lecture course given by the author in Berkeley during the 1963-64 academic year. Its concise, concentrated presentation makes the general structure theory of Lie groups far more accessible than before, making it a particularly welcome addition to the literature. Numerous exercises offer the reader the opportunity to gain a concrete and in-depth understanding of the material.

2.1. From the Introduction.

Although the usual curriculum of college mathematics begins with analytic geometry, most students will come into first contact with the rigorous foundations of this subject only in more advanced courses devoted to abstract linear algebra, general topology, or real variable theory. The connections between these genuinely mathematical disciplines and the topics treated in the beginners' courses are rarely spelled out in detail, but the serious student of mathematics must eventually re-examine the notions and facts presented to him in the beginning with respect to their mathematical content and validity.

It is the main purpose of this booklet to provide the student who is strongly motivated toward mathematics with an aid for passing through this stage. Specifically, what follows is an examination of the basic geometrical features of Euclidean three-space from the view of rigorous mathematics. This requires that even the simplest visually obvious facts concerning the relations between points, lines, and planes in space be rigorously deduced from the properties of the system of real numbers. For the future mathematician, the benefit to be derived from this enterprise is not merely a better understanding and mastery of geometry but actually the gaining of a foothold in the structure of general mathematical concepts and techniques. Indeed, our program here necessarily involves algebra and analysis as much as it involves geometry.

Chapter I discusses, in a partly informal way, the most basic mathematical material needed later on: the real number system, the rudimentary language of set theory, the abstract notion of a group, and the elementary facts concerning vector spaces. Chapter II deals with the Euclidean plane, including the foundations of trigonometry. Chapter III deals with Euclidean three-space. The main orientation is toward the determination and further analysis of the distance preserving transformations, because much of the elementary mathematics of general importance is encountered here.

The presentation is designed to enable and encourage the student to work his way through the material, rather than to give him all the necessary proofs in a completely detailed form ready for passive absorption. The exercises are meant to constitute an essential part of the text and make nonuniform demands on the reader, some requiring very little work, others requiring considerably more. The various concepts and techniques that have to be invoked are treated in a fashion as close as possible to the nature of the geometry to which they are applied here, and no attempt has been made to present them in their full generality. Hopefully, the restriction to the intuitively familiar environment of elementary geometry will facilitate the reader's entry into rigorous mathematics by focusing his attention on its use for directly visible purposes.

2.2. Review by: A E Maxwell.
The Mathematical Gazette 53 (384) (1969), 195-196.

This book, almost a pamphlet, is undoubtedly expensive in terms of volume, but it is interesting value for the money. After an opening chapter on basic algebraic requirements, the author proceeds to a study of the Euclidean plane via (among other things) inner products, complex numbers, and the exponential function for the circle. The third, and final, chapter, deals with Euclidean three-space: points, lines and planes; sides of planes, inner product and orthogonality; the quaternions; orthogonal linear maps.

This is a serious attempt to bring the language of geometry into that now current for algebra - an odd illustration, in some ways, of a process of commutativity. I found reading a little hard at times; but I am probably too deeply imbedded in the traditional language to be a fair test for the author. An interesting book.

2.3. Review by: C B Allendoerfer.
The American Mathematical Monthly 76 (3) (1969), 318.

The usual treatment of analytic geometry is based upon synthetic Euclidean geometry and proceeds without reference to the related subjects of linear algebra, group theory, and analysis. On the other hand, courses in these more advanced topics only rarely discuss applications of these topics to geometry. The purpose of this book is to fill this "geometry gap." It should be useful for students who wish to do transitional reading at this stage of their careers, but it can hardly be considered to be a text in the ordinary sense.

The content is elementary, but the exposition is sophisticated. The first chapter is a quick summary of real numbers, sets, maps, groups, and vector spaces. The second chapter treats the plane as a two-dimensional real vector space with an inner product. Lines are defined as translates of the one-dimensional vector subspaces. Rotations and reflections are special cases of distance preserving maps. The exponential function is rigorously introduced and is used to define sine and cosine and the measure of an angle.

The third chapter treats three-space in a similar fashion. A novel feature is the introduction of quaternions and their use to define the concepts of vector product and determinant.

The lack of an index prevents the author from giving an appropriate reference to Halmos. (Editor's Note: See the entry Hochschild in Finite Dimensional Vector Spaces (1958) by Halmos.)

3.1. From the Preface.

The theory of affine algebraic groups was originally developed in connection with Lie group theory, where these groups appeared as algebraic linear groups, i.e. as groups of automorphisms of finite-dimensional vector spaces, defined as subgroups of the full linear groups by algebraic conditions, such as the condition of fixing a given bilinear form. The first systematic and purely algebraic development of algebraic linear groups was carried out by Chevalley [C Chevalley, Theorie des groupes de Lie]. Concurrently with new developments in algebraic geometry, this theory was then built up to a theory of general algebraic groups, which contains the theory of algebraic linear groups as the special case of affine algebraic groups , i.e. groups whose underlying algebraic varieties are affine. Simultaneously, the structure theory of these affine groups was deepened and freed from the classical restriction to base fields of characteristic 0. A culmination of this development is [C Chevalley, Séminaire sur la classification des groupes de Lie algebriques].

Although the groups being investigated are affine, it soon becomes necessary to invoke much more than just affine algebraic geometry. The reason for this is that algebraic homogeneous spaces that are not affine algebraic varieties play a decisive role in the deeper structure theory of affine algebraic groups. A detailed account, ab initio, of the structure theory would therefore be excessively voluminous. For the student who is already equipped with algebraic geometry, the approach has recently been paved by Borel, in [A Borel, Linear Algebraic Groups (notes taken by H Bass)]. The present exposition has more modest aims. It is oriented toward representation theory, and its principal aim is to fuse representation-theoretical technique with the elementary theory of affine algebraic groups so as to provide an efficient tool, especially for Lie group theory. The main material covered is in essence, though not in form, that of [C Chevalley, Theorie des groupes de Lie]. Considerable emphasis is placed on the linearisation technique based on Lie algebras, which becomes effective only when the base field is assumed to be of characteristic 0.

The setting in which we operate arises as follows. A finite-dimensional representation theory for any group $G$ is based on the choice of a category of admissible $G$-modules, and hence on the choice of an associated family of representative functions on $G$. The family of these functions has the structure of a Hopf algebra over the chosen base field. The comultiplication of this Hopf algebra reflects the group multiplication of $G$, while the multiplication of the Hopf algebra (which is simply the value-wise multiplication of functions) reflects the diagonal map $G \rightarrow G \times G$, and corresponds to the formation of tensor products of $G$-modules. The $G$-modules under investigation may be viewed as the comodules for the Hopf algebra of representative functions. On the other hand, this Hopf algebra defines a group $G^{*}:$ namely, the group of all finite specialisations of the Hopf algebra into the base field. This group $G^{*}$ may be regarded as a universal algebraic hull of $G$. It has the structure of a pro-affine algebraic group i.e. a projective limit of affine algebraic groups), and the representation theory of $G$ becomes the "rational" representation theory of $G^{*}$ as a pro-affine algebraic group.

The importance of the theory of affine algebraic groups for general representation theory is immediately evident from this. The passage from affine algebraic groups to pro-affine algebraic groups is simply the passage from finitely generated Hopf algebras to general ones, and every Hopf algebra is the union of the family of its finitely generated Hopf subalgebras. Once the affine theory has been brought under control, the extension to the pro-affine theory is not difficult. Here, we remain with the affine situation.

In order to keep this book self-contained, we have limited the material so that only elementary affine algebraic geometry comes into play, and all the results in this area that we use are established in Section 1. All that is presupposed with regard to the elementary theory of Lie algebras, used in the later sections, will be found in Chapters X and XI of [G Hochschild, The Structure of Lie Groups] or, of course, in [N Jacobson, Lie Algebras]. Otherwise, no special knowledge is presupposed, although it is assumed that the reader will be able to assimilate the daily diet of the working algebraist in unpremasticated form. The organisation should be clear from the section headings.

3.2. Review by: James E Humphreys.
Mathematical Reviews MR0277535 (43 #3268).

This book offers a concise, attractive reformulation, from the viewpoint of Hopf algebras of representative functions, of the material in C Chevalley's Théorie des groupes de Lie. Tome II: Groupes algébriques [1951], and of related topics. Because heavy use is made of the Lie algebra, the deeper results are obtained only for characteristic 0 but this is adequate for the applications to Lie group theory that the author outlines in his preface. For the general theory of linear algebraic groups, the reader should turn instead to A Borel's Linear algebraic groups [1969].

Only affine algebraic geometry is used here, and this appears in the guise of commutative algebra. Aside from a knowledge of Lie algebras, the text presupposes only standard algebra. The style is that of the author's in The structure of Lie groups [1965] (though not quite so relentless).

An "affine algebraic group" $G$ over an arbitrary field $F$ is given by a finitely generated Hopf algebra $A$ (with antipode) over $F$, which consists of representative functions on $F$. A rough outline of topics covered in the text is as follows: identity component of $G$, image of $G$ under a morphism, factor groups, Lie algebra of $G$, subgroup-subalgebra correspondence (characteristic 0), polynomial (i.e., rational) representations of $G$ and related ideals in the Lie algebra, unipotent radical of $G$, structure of abelian groups, Lie-Kolchin theorem, algebraic Lie algebras (characteristic 0), Levi decomposition (characteristic 0), structure of $Aut G$. There are useful exercises, giving examples, counterexamples (in characteristic $p$), and complements.

At times the emphasis on the algebra $A$ seems (to the reviewer) to detract from the geometric content of the elementary results (cf. the rather elaborate discussion of the identity component, pp. 26-27). Nonetheless, the formalism of Hopf algebras does provide a coherent alternative to Chevalley's original treatment, which the reviewer has always found difficult to penetrate.

4.1. From the Publisher.

The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras. It is thus an ideally suitable framework for exhibiting basic algebra in action. To do that is the principal concern of this text. Accordingly, its emphasis is on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of the simple groups and their irreducible representations. In the same spirit, this exposition has been made entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years of graduate study in algebra is presupposed.

The chapter headings should be sufficient indication of the content and organisation of this book. Each chapter begins with a brief announcement of its results and ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references. The references are intended to be merely suggestions for supplementary reading or indications of original sources, especially in cases where these might not be the expected ones. Algebraic group theory has reached a state of maturity and perfection where it may no longer be necessary to re-iterate an account of its genesis. Of the material to be presented here, including much of the basic support, the major portion is due to Claude Chevalley.

4.2. Review by: R H Dye.
Mathematical Reviews MR0620024 (82i:20002).

The author's aim is clearly expressed in his preface. He says: "The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras. It is thus an ideally suitable framework for exhibiting algebra in action. To do that is the principal concern of this text. Accordingly, its emphasis is on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of the simple groups and their irreducible representations. In the same spirit, this exposition has been made entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years of graduate study in algebra is presupposed." The chapter headings are well chosen to indicate the content and approach. They are: Representative functions and Hopf algebras; Affine algebraic sets and groups; Derivations and Lie algebras; Lie algebras and algebraic subgroups; Semisimplicity and unipotency; Solvable groups; Elementary Lie algebra theory; Structure theory on characteristic 0; Algebraic varieties; Local theory; Coset varieties; Borel subgroups; Applications of Galois cohomology; Algebraic automorphism groups; The universal enveloping algebra; Semisimple Lie algebras; From Lie algebras to groups. The reviewer enjoyed reading the book, learned from it, and welcomes this addition to the mathematical literature. In his view this book achieves its aims.

5.1. From the Preface.

Primarily, this book addresses beginning graduate students expecting to become mathematicians or mathematically oriented computer scientists. Accordingly, the presentation is conditioned in content as well as in form by the assumption that the reader has already made an internal commitment to mathematics and is seeking not only mathematical information but also active involvement with mathematical pursuits.

The general aim of what follows is to present basic mathematical concepts and techniques in familiar contexts in such a way as to illuminate the nature of mathematics as an art. Thus, the selection and organisation of the material is based on considerations regarding the philosophical significance of various mathematical notions and results, their interdependence and their accessibility. In other words, this text concentrates on displaying coherent mathematical material meriting exceptionally careful and expansive contemplation. It should not be regarded as a reference for the most frequently used results or methods of elementary mathematics.

The exposition is meant to be reasonably self-contained and to embody a growth pattern of mathematical ideas (for which no historical validity is claimed, of course). In order to avoid burying the essentials under routine technicalities, a style has been adopted that relies on the reader's active involvement somewhat more than is customary in texts for beginners. The exercises proposed at the end of each chapter are frequently extensions of the chapter content, rather than mere illustrations. They are designed to be manageable in a straightforward fashion within the framework provided by the text.

The projects following the exercises are concerned with the creation of computer programs. However, for the purposes at hand, it is not necessary to do any formal programming. It suffices to think of a computer program in the informal sense, as a detailed recipe for accomplishing a computational task by a routine that, in principle, can be completely automated. The point to be made by the inclusion of such projects is that often the construction of a formal proof in a mathematical exposition is essentially the same enterprise as the creation of a computer program. At any rate, it is hoped that, in the contexts to which the projects refer, the programming enterprise will contribute to the clarification and appreciation of the underlying mathematics. Moreover, the reader with some programming expertise may find that the complete creation and running of the suggested programs is intellectually rewarding.

5.2. Chapter headings.

I. Counting;
II. Integers;
III. Fractions;
IV. Linearity;
V. Multilinear algebra;
VI. Polynomials;
VII. The exponential function;
VIII. Integration;
IX. The sphere in 3-space.

5.3. Review by: John Baylis.
The Mathematical Gazette 69 (447) (1985), 72-73.

This book, for beginning graduate students, aims to "present basic mathematical concepts and techniques in familiar contexts in such a way as to illuminate the nature of mathematics as an art". It is a short book and the selection of material has been controlled by the "philosophical significance of various mathematical notions and results, their interdependence and their accessibility". This is clearly a difficult aim to achieve and I interpreted it as meaning that the book would provide an introduction to how mathematicians think about and do mathematics, via a few very main stream topics.

To criticise the book in terms of what is left out is therefore unfair, so let us look at what is in it. There are nine chapters. The first four certainly sound very basic: Counting, Integers, Fractions, Linearity. "Counting" consists of nine pages, seven of which are used up telling our graduate student what a set is, the meanings of $\cap, \cup, \subset$, associative, mapping, maximal, $\binom{n}{k}$, etc. The only meat of the chapter is in a very nice but subtle proof of the marriage theorem. It just made me wonder: what sort of graduate student needs to be treated like this?

The integers then take up eleven pages and from the outset are rooted firmly in the context of Abelian groups. The chapter is mainly an excuse to talk about the terminology and standard results in groups, rings, fields and their homomorphisms. Setting objects which we all know and love, like the integers, into a wider context can of course be enlightening and valuable, but the student who doesn't already know all this stuff is certainly going to wonder if the essence of pure mathematics consists in the invention of higher and higher technology to crack peanuts.
...

The content of Chapter 3 is fractions, decimals, Cauchy sequences of rationals, continued fractions, and Liouville's result on rational approximations to zeros of rational polynomials. There is some very nice mathematics in the discussion of the lengths of the periodic and non-periodic parts of recurring decimal expressions and of the periodicity of quadratic continued fractions - nice largely because it is within the scope of a first year undergraduate or good sixth-former, and most of the proofs are carried out at that level. But for some reason the author seems not content with this and the reader is made to feel that he is not doing things properly unless he thinks in terms of subrings of the rationals, the order of elements in the group of units in $\mathbb{Z}/\mathbb{Z}\nu$, negligible Cauchy sequences as an ideal in the ring of all Cauchy sequences, etc. Important as these ideas are in "higher" mathematics, they are not used in any significant way here and we have yet another deployment of cannons to shoot flies. At the start of this chapter we find: "One verifies directly that there is one and only one way of making the set of all fractions into a field $\mathbb{Q}$ such that the map sending each integer $x$ to the fraction $x/1$ is an injective ring homomorphism". What is the student who needs such a lengthy explanation of union and intersection to make of that?

The remaining chapters are Linearity, Multilinear algebra, Polynomials, The exponential function, Integration and The sphere in 3-space. The pace is much hotter in these chapters and the "higher" point of view is used significantly to establish important results. However, the outline proofs are so brief that even a relatively experienced graduate student would have to commit himself to several weeks of hard work to fill the gaps. Well, the preface did say the book was for committed students, and there is nothing wrong with hard work, but I could find nothing in the book which could be called motivation or encouragement, and there are many books which make hard work much more attractive.

To end on a rather more positive note, the book does have, as well as the (mainly hard) exercises a selection of "projects" for each chapter. These are to devise computer programs (or detailed algorithms - since the reader is not assumed to have done any computing) for bringing some rather abstractly treated ideas down to earth, for example, to generate the continued fraction expansion of a quadratic irrational, to carry out quaternion arithmetic, to determine winding indices, to produce all partitions of $n$ into $s$ summands, to solve the marriage problem recursively, etc.