Karl H. Hofmann and Sidney A. Morris

The Structure of Compact Groups
A Primer for the Student - A Handbook for the Expert

de Gruyter Studies in Mathematics, Volume 25
1998. 17 x 24 cm. xvii, 835 pages. ISBN 3-11-015268-1
Second Revised and Augmented Edition, August 2006.
XVII, 858 pages, ISBN 978-3-11-019006-9 (ISBN10: 3-11-019006-0)

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Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups.

After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. However, the thrust of book points in the direction of the structure theory of infinite dimensional, not necessarily commutative compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups.

The first edition of 1998 was well received by reviewers and has been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and some minor inaccuracies of content; it has been edited and improved in various sections. New material has been added in order to reflect ongoing research. In the process of revising the original edition, the integrity of the original section numbering was carefully respected so that citations of material from the first edition remains perfectly viable to the users of this edition.

A review of the first edition of this book which appeared in the June 1999 issue, Volume 32, Page 33 of the European Mathematical Society Newsletter follows:

"This is, from several points of view, a very impressive book which aspires to become an important monograph, a reference book as well as a textbook for more advanced students. 615 pages of the main text, plus 220 pages devoted to four appendices (the index takes 37 pages), show the extent of the material the book covers. Further, it is clear from the organization of the book that the authors have much experience in presenting the subject. They have found ways of making a reader quickly familiar with parts of the theory, without requiring a lot of preliminaries. This will doubtless stimulate further, more difficult, readings in the book. There are many interesting examples and exercises (with hints if necessary) which will substantially improve the knowledge of a reader and which make the text more attractive.

At the beginning of each chapter the reader is told which prerequisites are needed and where to find them, and at the end there are references for additional reading. The authors have included many remarks, comments, and postscripts at the end of each chapter to clarify the main text, and have tried to make the book as self-contained as possible. For this purpose they have included appendices on abelian groups, covering spaces and groups, a primer on category, and selected results in topology and topological groups.

The book deals with compact topological groups, written more from the point of view of general topology than from that of algebraic topology. Nevertheless, much information about algebraic topology and homological properties of compact groups, can be found. The representation theory of compact groups is presented from a general point of view, so we cannot expect to find lists of representations of all the classical groups. Lie groups play a very important role in the whole book, but this is not a textbook on Lie groups. Not only does the book present a lot of material, but there are many results that have previously appeared only in articles. The book will probably also be useful to non-specialists in the field. If one is looking for a notion or a result, one can find and understand it without reading the book systematically from the very beginning. We expect that the book will be on the shelves of many mathematicians, as well as many students of mathematics."

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