Topological Groups: Yesterday, Today, Tomorrow
Edited by Sidney A. Morris
MDPI (Basel, Switzerland)
vii+217pp, 2016, ISBN 978-03842-268-6
In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book "Hilbert's Fifth Problem and Related Topics" by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao.
It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 "The Structure of Compact Groups" by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and Pavel Zalesskii (2012). The 2007 book "The Lie Theory of Connected Pro-Lie Groups" by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups.
The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangel?ski? and many of his former students who developed this topic and its relations with topology. The book "Topological Groups and Related Structures" by Alexander Arkhangel?skii and Mikhail Tkachenko has a diverse content including much material on free topological groups.
Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day.
The Lie Theory of Connected Pro-Lie Groups|
A Structure Theory for Pro-Lie Algebras,
Pro-Lie Groups, and Connected Locally
European Mathematical Society Publishing House
xv+678pp, May 2007, ISBN 978-3-03719-032-6
For further publication information see the Publisher's website
Review in Jahresbericht der Deutschen Mathematiker 111(2009)
Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie.
A century later Jean Dieudonné quipped that Lie groups had moved to the center
of mathematics and that one cannot undertake anything without them.
If a complete topological group G can be approximated by Lie groups in the sense
that every identity neighborhood U of G contains a normal subgroup N such that G/N
is a Lie group, then it is called a pro-Lie group. Every locally compact connected
topological group and every compact group is a pro-Lie group. While the class of locally
compact groups is not closed under the formation of arbitrary products, the class of pro-Lie
For half a century, locally compact pro-Lie groups have drifted through the literature,
yet this is the first book which systematically treats the Lie and structure theory of
pro-Lie groups irrespective of local compactness. This study fits very well into that
current trend which addresses infinite dimensional Lie groups. The results of this text
are based on a theory of pro-Lie algebras which parallels the structure theory of finite
dimensional real Lie algebras to an astonishing degree even though it has to overcome
greater technical obstacles.
This book exposes a Lie theory of connected pro-Lie groups (and hence of connected
locally compact groups) and illuminates the manifold ways in which their structure
theory reduces to that of compact groups on the one hand and of finite dimensional
Lie groups on the other. It is a continuation of the authors' fundamental monograph
on the structure of compact groups (1998, 2006), and is an invaluable tool for researchers
in topological groups, Lie theory, harmonic analysis and representation theory. It
is written to be accessible to advanced graduate students wishing to study this
fascinating and important area of current research, which has so many fruitful
interactions with other fields of mathematics.
The Structure of Compact Groups|
Primer for the Student -
A Handbook for the Expert
3rd Edition, 2013
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.
For further publication information see Publisher's website
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.
Errata and Addenda to 3rd Edition
Errata and Addenda to the previous edition
Pontryagin Duality and the Structure of|
Algebra and |
| Sidney A. Morris |
Pontryagin Duality and the Structure of
Locally Compact Abelian
Cambridge University Press
1977, 136 pp.
ISBN 0 521 21543 9
The book was translated
and published by
pdf file of book
Review of book
A review of this
book which appeared in Acta Scientarium Mathematicarum
"One of the central results in the theory of locally compact abelian groups
is the Pontryagin-van Kampen duality theorem which implies that a locally
compact abelian group is completely determined by its dual and thus yields
a powerful method to study the structure of such groups. Utilizing this
fact, the author gives an approach to the structure theory of locally
compact abelian groups which proceeds simultaneously with the derivation of
the duality theorem. This approach is made possible by a new and simple
proof of the duality theorem, which beyond some basic facts from group
theory and topology, presupposes only the Peter-Weyl theorem.
First, a concise general introduction to the theory of topological groups,
some basic facts concerning subgroups and quotient groups of Rn and
concerning uniform spaces are given. Then dual groups are introduced.
The duality theorem is proved first for compact and discrete abelian
groups and then extended to all locally compact abelian groups. The
structure theory of locally compact abelian groups including the Principal
Structure Theorem is derived simultaneously. Then some consequences of
the duality theorem and applications in diophantine approximations are
discussed. The structure theory is further developed by considering its
relations to the structure theory of general locally compact groups. At last
some important results are given concerning the structure of non-abelian
locally compact groups. Each chapter contains a number of stimulating
and illustrating exercises, which help to develop the reader's technique.
The author's skill and exceptional knowledge of the subject enabled him to
achieve his purpose completely. The lecture note is very clearly and
elegantly written and can be recommended as a text for first year
graduate courses both by its content and by the educational value of its
J. Szenthe (Budapest)"
| Arthur Jones,
Sidney A. Morris,|
and Kenneth R. Pearson
Abstract Algebra and Famous Impossibilities
New York, Berlin etc.
187 pp. 27 figs., Softcover
1st ed. 1991.
Corr. 2nd printing 1993.
Review of book
| The famous problems of
squaring the circle, doubling the cube, and trisecting
the angle have captured the imagination of both
professional and amateur mathematician for over two thousand years.
These problems, however, have not yielded to purely geometrical methods.
It was only the development of abstract algebra in the nineteenth century
which enabled mathematicians to arrive at the surprising conclusion that
these constructions are not possible. This text develops the abstract
algebra and geometry necessary to prove that these constructions are
impossible. It is written at a level suitable for students who have already
taken a course in linear algebra and contains thorough discussion of the
new concepts along with many illustrative examples. A large number of
carefully graded examples are included. |
A review of this
book which appeared in the Australian Mathematical Society
This book gives complete proofs that it is not possible,
using only straight edge and compass, to trisect an arbitrary angle, double
a cube or square a circle. It is intended to be used as a text at about
second year level.
The proofs that these geometric constructions are impossible make a good
framework for a second year course. The impossibility assertions provide
a clear objective which can be stated and understood with only a
rudimentary knowledge of geometry. Along the way to this objective
students learn some quite sophisticated algebra and calculus.
The authors have taught this topic over a number of years and the benefit
of that experience is evident in the text. It is written clearly and great care
is taken to explain what each chapter, section, theorem and step is about
and how they all fit together to form the proofs of the impossibilities. The
material is, necessarily, abstract and involves many proofs but the
authors present it in a way which ought to make it accessible to most
students. There are exercises after each section for the students to
practise what they have just read about. For those students (or
instructors) who want to know more, there is at the end of each chapter an
inviting list of ‘Additional Reading.
The first chapter is called "Algebraic Preliminaries" and it summarises the
basics of fields, rings and vector spaces. Students are assumed to be
already familiar with fields and vector spaces. Straight edge and compass
constructions are then defined and the proofs of the impossibilities
completed. Knowledge of basic integration theory, but not complex
analysis, is assumed for the proof that pi is transcendental.
This linking, and solving, of problems in geometry, calculus, and the
theory of equations, even though they are at first sight quite unrelated,
illustrates very well the power of abstract thinking and the unity of
mathematics. That relatively modern techniques were required to solve
ancient problems shows the historical scope of mathematics. We should
be showing students such things and ‘Abstract Algebra and Famous
Impossibilities’ does that very clearly.
George Willis (Newcastle)
Updated November 19, 2013
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