Karl H. Hofmann

The subject matter of compact groups is frequently cited in fi elds like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a text for upper level graduate students, and of being a source book for researchers who need 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 and on locally compact abelian groups. Appended chapters contain the material for self-contained courses on abelian groups and on category theory. Using the Lie algebras and the exponential function of arbitrary compact groups, the book avoids unnecessary restrictions to finite dimensional or abelian compact groups. Earlier editions of 1998, 2006, 2013, and 2020 have been quoted for instruction and research. The present edition conceptually sharpens, polishes, and improves the earlier material. For instance, it includes a treatment of the Bohr compactifi cation of topological groups which fi ts perfectly into the general treatment of adjoint functors that the book treats in an appendix of its own, and which, in the abelian environment, connects neatly with the Pontryagin--van Kampen duality of compact abelian groups having been discussed in the book in great detail. The link between arbitrary compact groups and their weakly complete group algebras is as extensively discussed as is now the theory of weakly complete universal enveloping algebras of the Lie algebras of compact groups. All of this is based on the category of weakly complete real and complex vector spaces and its precise duality to the category of ordinary real, respectively, complex vector spaces, is treated in an appendix systematically.

This authoritative book on periodic locally compact groups is divided into three parts: The first part covers the necessary background material on locally compact groups including the Chabauty topology on the space of closed subgroups of a locally compact group, its Sylow theory, and the introduction, classifi cation and use of inductively monothetic groups. The second part develops a general structure theory of locally compact near abelian groups, pointing out some of its connections with number theory and graph theory and illustrating it by a large exhibit of examples. Finally, the third part uses this theory for a complete, enlarged and novel presentation of Mukhin’s pioneering work generalizing to locally compact groups Iwasawa’s early investigations of the lattice of subgroups of abstract groups. Contents Part I: Background information on locally compact groups Locally compact spaces and groups Periodic locally compact groups and their Sylow theory Abelian periodic groups Scalar automorphisms and the mastergraph Inductively monothetic groups Part II: Near abelian groups The definition of near abelian groups Important consequences of the definitions Trivial near abelian groups The class of near abelian groups The Sylow structure of periodic nontrivial near abelian groups and their prime graphs A list of examples Part III: Applications Classifying topologically quasihamiltonian groups Locally compact groups with a modular subgroup lattice Strongly topologically quasihamiltonian groups

Of all topological algebraic structures compact topological groups have perhaps the richest theory since 80 many different fields contribute to their study: Analysis enters through the representation theory and harmonic analysis; differential geo­ metry, the theory of real analytic functions and the theory of differential equations come into the play via Lie group theory; point set topology is used in describing the local geometric structure of compact groups via limit spaces; global topology and the theory of manifolds again playa role through Lie group theory; and, of course, algebra enters through the cohomology and homology theory. A particularly well understood subclass of compact groups is the class of com­ pact abelian groups. An added element of elegance is the duality theory, which states that the category of compact abelian groups is completely equivalent to the category of (discrete) abelian groups with all arrows reversed. This allows for a virtually complete algebraisation of any question concerning compact abelian groups. The subclass of compact abelian groups is not so special within the category of compact. groups as it may seem at first glance. As is very well known, the local geometric structure of a compact group may be extremely complicated, but all local complication happens to be "abelian". Indeed, via the duality theory, the complication in compact connected groups is faithfully reflected in the theory of torsion free discrete abelian groups whose notorious complexity has resisted all efforts of complete classification in ranks greater than two.