The Structure of Compact Groups

Karl H. Hofmann, Sidney A. Morris

• Publisher: De Gruyter
• Year: 2013
• Print ISBN: 9783110296556
• eBook ISBN: 9783110296792
• Edition: 3
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on it 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. Separate appended chapters contain the material for courses on abelian groups and on category theory. However, the thrust of the book points in the direction of the structure theory of not necessarily finite dimensional, nor 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 and the second edition of 2006 were well received by reviewers and have been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and the content has been conceptually sharpened in some places and polished and improved in others. New material has been added to various sections taking into account the progress of research on compact groups both by the authors and other writers. Motivation was provided, among other things, by questions about the structure of compact groups put to the authors by readers through the years following the earlier editions. Accordingly, the authors wished to clarify some aspects of the book which they felt needed improvement. The list of references has increased as the authors included recent publications pertinent to the content of the book.

Table of contents

• Chapter 1. Basic Topics and Examples
• Chapter 2. The Basic Representation Theory of Compact Groups
• Chapter 3. The Ideas of Peter and Weyl
• Chapter 4. Characters
• Chapter 5. Linear Lie Groups
• Chapter 6. Compact Lie Groups
• Chapter 7. Duality of Abelian Topological Groups
• Chapter 8. Compact Abelian Groups
• Chapter 9. The Structure of Compact Groups
• Chapter 10. Compact Group Actions
• Chapter 11. The Structure of Free Compact Groups
• Chapter 12. Cardinal Invariants of Compact Groups
• Appendix 1. Abelian Groups
• Appendix 2. Covering Spaces and Groups
• Appendix 3. A Primer of Category Theory
• Appendix 4. Selected Results on Topology and Topological Groups
• Appendix 5. Measures on Compact Groups
• Appendix 6. Projective Limits of Well-Ordered Inverse Systems
• References
• Index of Symbols
• Index