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The structure of compact groups. A primer for the student – a handbook for the expert. 2nd revised and augmented edition. (English) Zbl 1139.22001

de Gruyter Studies in Mathematics 25. Berlin: Walter de Gruyter (ISBN 3-11-019006-0/hbk). xvii, 858 p. (2006).
This is the second edition of the book, which is revised and augmented in several parts. For a detailed review of the first edition (1998), which in particular includes the contents, see Zbl 0919.22001. The structure of the book does not change up to the level of sections; some augmentations appear as additional subsections at the ends of the existing sections. In two cases, the additional portions of information are marked by “bis” (Theorem 8.36bis and Theorem 9.76bis) but usually they are added to the existing propositions or theorems. The numbering system of the first edition does not change (i.e., the references to the first edition remain valid). The list of references contains 285 items, including 20 new ones. A list of errata to the first edition is accessible on the home page of the second author.
The main theme of the book is formulated in its title: “The Structure of Compact Groups”. Thus, the authors do not consider systematically applications of the theory; related fields (for example, representations) are considered if they clarify the structure of compact groups.
In the preface to the first edition, the authors gave a clear description of the leading ideas of the book. The content of the book is described as follows: “In a simplified fashion one might say that this is, in the first place, a book on the structure of connected compact groups and, in the second, on the various ways that general compact groups are composed of connected compact groups and of totally disconnected ones”. Thus, the emphasis is on the non-Lie case. Roughly, the book covers the non-smooth part of the compact groups theory, except for special problems on profinite groups. This choice is quite reasonable. Indeed, many working mathematicians (and physicists as well) use the machinery of compact Lie groups and related fields – root systems, Young diagrams, Jacobi and other orthogonal polynomials, flag manifolds, the Weyl integration formula, the Kirillov formula, et cetera. As a consequence, there is plenty of excellent monographs which include the information on compact Lie groups as an important necessary portion (usually, the exposition is adapted to the main theme of a book – harmonic analysis, algebraic groups, symmetric spaces, Lie algebras, representation theory, \(C^*\)-algebras, mathematical physics, etc.). This book has relatively small overlaps with them; on the other hand, it contains many useful facts which were contained only in research papers (for example, results on free compact groups).
The following quotations characterize the authors’ approach: “In many respects, in the end, one might consider our approach as a general Lie theory for compact groups, irrespective of dimension”; “It is generally believed that the approximation of arbitrary compact groups by Lie groups settles an issue on the structure of the compact groups as soon as it is resolved for Lie groups. There is veracity to this legend, as most legends are founded in reality – somewhere, but this myth is far from reflecting the whole truth”. Many mathematicians who work with compact Lie groups think that any problem on general compact groups can be reduced to the Lie group case using the approximation by quotients over small normal subgroups. However, even if this is possible, the procedure and the answer are not always evident. Many well-known facts and notions of compact Lie groups theory do not admit an obvious generalization. For example, let \(G\) be a connected compact group. What is (or could be, or should be) a Lie algebra of \(G\)? A fundamental group of \(G\)? A universal covering group of \(G\)? Note that the standard theory does not work since there exist connected compact groups which are not locally arcwise connected (see Theorem 8.36, (\(\roman 2\))). The book contains answers to the questions above.
The authors’ approach permits to give a self-contained exposition and to achieve the aim that is formulated in the sub-title of the book: “A Primer for the Student – a Handbook for the Expert”. To write a book that realizes both of the two parts of the sub-title is a highly nontrivial problem but the authors found a solution for it. Chapter 9, which has the same title as the book, is the heart of the book. The preceding chapters prepare the necessary material for the structure theorems; together with the appendices they may be treated as a Primer (however, Chapter 7 and Chapter 8, which are devoted to the duality theory of abelian topological groups and to the structure of compact abelian groups, respectively, are not quite elementary). Chapters {10–12} contain material for experts (compact group actions, free compact groups, cardinal invariants). The book is carefully written. The exposition is detailed and clear. There are many exercises. Usually, they are supplied with hints, which are more like sketches of proofs sometimes. All chapters, including the appendices, finish with sections “Postscript”, which briefly summarize their content, and “References for Chapter – Additional reading”. Due to the detailed “Index” and “List of Symbols”, the reader can easily find further information. The non-standard approach requires a non-standard exposition. Since the book is a result of 20–30 year classrooms and lectures, the innovations are tested on several generations of students. Perhaps, the main feature of the exposition is the definition of a compact Lie group as a compact linear group; this makes it possible to avoid difficulties of the standard approach (smooth manifolds, vector fields, etc.). A monograph of almost 900 pages ({\(\roman 27\)}+858), of course, cannot be free of shortcomings; I noted a few: in the statement of Theorem 9.32, one has to add that a subgroup is abelian to the last phrase of ({\romannumeral5}); in the statement of Theorem 9.41, the assertion that \(D\cap G_0\subseteq Z(G_0)\) is repeated twice (p. 479); the list of references does not contain essential papers of V. Berestovskii and C. Plaut on the covering group theory [J. Pure Appl. Algebra 161, No. 3, 255–267 (2001; Zbl 0996.22006); Topology Appl. 114, No. 2, 187–199 (2001; Zbl 0982.22003); Topology Appl. 114, No. 2, 141–186 (2001; Zbl 0982.22002)]; there is a new word action given on page 59.
Thus, the book is informative and carefully written; the exposition is fresh and excellent. It can be useful for students as well as for experts and will meet, as well as the first edition, a wide audience.

MSC:

22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22C05 Compact groups
22B05 General properties and structure of LCA groups
22E15 General properties and structure of real Lie groups
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
54H11 Topological groups (topological aspects)
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