Table Of ContentAn Introduction to
Groups, Groupoids and
Their Representations
Alberto Ibort
Department of Mathematics
University Carlos III of Madrid
Madrid, Spain
Miguel A Rodrı´guez
Department of Theoretical Physics
University Complutense of Madrid
Madrid, Spain
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A SCIENCE PUBLISHERS BOOK
A SCIENCE PUBLISHERS BOOK
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To our wives, Conchi and Teresa
and our sons, Alberto and Pedro
Preface
This project grew from lecture notes prepared for courses on group theory taught
by the authors a long time ago. They were recovered and put in their present
form because of the impetus provided by recent investigations on groupoids
that convinced the authors of the interest in providing a way to graduate and
undergraduate students as well to researchers without a strong background on
geometry and functional analysis to access the subject.
The theory of groups and their representations has had an extraordinary
success and reached a state of maturity in the XXth century. Motivated by problems
in Physics, mainly Quantum Mechanics, the theory of group representations had
a fast development in the first half of the century, boosted by the contributions
and insight of such extraordinary people as H Weyl, E Wigner, etc. Later on, the
definitive contributions of A Kirillov, B Kostant, GW Mackey and many others
closed the circle by establishing the deep geometrical link between the theory of
group representations and symplectic geometry and quantization. There are many
books on the subject covering the full spectrum of rigor, depth and scope: From
elementary to advanced, encyclopedic and highly specialized, etc.
At the same time, the notion of groupoid [7], the ‘most natural’ extension
of the notion of group, sometimes even proposed to be called ‘groups’, has
had a very different history. A Connes claimed that “it is fashionable among
mathematicians to despise groupoids and to consider that only groups have
an authentic mathematical status, probably because of the pejorative suffix
oid”, but that they are the relevant structures behind many notions in Physics
[17, page 13].
Whatever the reason, the truth is that the theory of groupoids has not been
developed to the extent that the theory of groups has. Only more recently, because
of various facts, among them the insight coming from Operator algebras (see
for instance the book by Renault [57]); the relevant role assigned to them in
the mathematical foundations of classical and quantum mechanics in the work
by N Landsman [45]; the combinatorial applications in graph theory [43]; the
applications of groupoid theory to geometrical mechanics, control theory (see,
vi < An Introduction to Groups, Groupoids and Their Representations
for instance, [19] and references therein) and variational calculus [52]; the recent
contributions to the spectral theory of aperiodic systems [4], the recent groupoids-
based analysis of Schwinger’s algebraic foundation for Quantum Mechanics [14],
[15], or the appealing description by A Weinstein of groupoids as unifying internal
and external symmetries [66] extending, in a natural way, the notion of symmetry
groups and groups of transformations with the associated wealth of applications,
groupoids are attracting more and more attention.
There are many treatments of groupoid theory in the literature, from the
very abstract, like the seminal categorical treatment in [33], to the differential
geometrical approach in [48], passing through the Harmonic analysis perspective
in [67] or the C*-algebraic setting [57] mentioned already. The elementary notions
on groupoids can be explained, however, at the same level that those of groups, and
actually, some of the ideas on group theory related to groups as transformations
on spaces, gain in clarity and naturality when presented from the point of view of
groupoids. We feel that there is a lack of a presentation, as elementary as possible,
of the theory and structure of the simplest possible situation, that is, of finite
groupoids, that will help researchers, in many areas where group theory is useful,
to effectively use groupoids and their representations.
On the other hand, the theory of linear representations of finite groupoids
is interesting enough to deserve a detailed study by itself. We will just mention
here that the fundamental representation of the groupoid of pairs of n elements
is just the matrix algebra M(C). Hence, the theory of linear representations
n
of finite groupoids provides a different perspective to the standard algebra of
matrices. There are a number of relevant contributions to the theory of linear
representations of groupoids that describe the notion of measurable and continuous
representations (see, for instance, the aforementioned work by Landsman [45] or
[6] and references therein), unitary representations of groupoids and the extension
of Mackey’s imprimitivity theorem [54], [55], [29] or the representation theory of
Lie groupoids [30], but again, as before we feel that there is lack of an elementary
approach to the theory of representations of groupoids discussing the theory for
finite groupoids.
Hence the proposal to write a monograph developing the foundations of
groups and groupoids simultaneously and at the same elementary level, enriching
both approaches by explaining the basic concepts together and providing a wealth
of examples. The first part of this work will describe the algebraic theory and
try to dwell in the structure of finite groups and groupoids. The second part will
develop the theory of linear representations of finite groups and groupoids.
There are many subjects that will not be covered in this work, like the theory of
topological and Lie groups and groupoids (the latter being particularly interesting
because of the recent proof of Lie’s third theorem for groupoids [20], [21]) or the
many applications mentioned earlier that will be left for further developments. We
believe that the time is ripe for a project like this, so we will try to offer the reader
a pleasant and smooth introduction to the subject.
Acknowledgements
Of course, this work would not have been possible without the help and complicity
of many people, starting with the closest ones, our families, thanks for your infinite
patience and support, to our colleagues, students and friends.
As mentioned above, this text has profited from old lecture notes on group
theory. Thanks to JF Carin˜ena for even older notes that inspired many parts of
this text.
We would also like to thank many of our teachers: LJ Boya, JF Carin˜ena,
M Lorente, from whom we got infected since our Ph.D. studies with the virus of
the harmony and inner beauty of group theory that has evolved into the present
book.
During the period of preparation of the manuscript, various aspects of the
book were discussed with many people in different contexts and presentations.
In particular AI would like to thank the students of the Master course on Abstract
Control Theory at the Univ. Carlos III de Madrid that have ‘suffered’ the exposition
of parts of this work.
Finally we would like to thank the editor of this book for his patience and
encouragement.
Contents
Preface v
Acknowledgements vii
Introduction xv
PART I: WORKING WITH CATEGORIES
AND GROUPOIDS
1. Categories: Basic Notions and Examples 3
1.1 Introducing the main characters 3
1.1.1 Connecting dots, graphs and quivers 3
1.1.2 Drawing simple quivers and categories 6
1.1.3 Relations, inverses, and some interesting problems 8
1.2 Categories: Formal definitions 10
1.2.1 Finite categories 10
1.2.2 Abstract categories 12
1.3 A categorical definition of groupoids and groups 15
1.4 Historical notes and additional comments 17
1.4.1 Groupoids: A short history 17
1.4.2 Categories 17
1.4.3 Groupoids and physics 17
1.4.4 Groupoids and other points of view 18
2. Groups 19
2.1 Groups, subgroups and normal subgroups: Basic notions 20
2.1.1 Groups: Definitions and examples 20
2.1.2 Subgroups and cosets 22
2.1.3 Normal subgroups 26
