1st Edition

An Introduction to Groups, Groupoids and Their Representations

ISBN 9781032086767
Published June 30, 2021 by CRC Press
362 Pages

USD $54.95

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Book Description

This book offers an introduction to the theory of groupoids and their representations encompassing the standard theory of groups. Using a categorical language, developed from simple examples, the theory of finite groupoids is shown to knit neatly with that of groups and their structure as well as that of their representations is described. The book comprises numerous examples and applications, including well-known games and puzzles, databases and physics applications. Key concepts have been presented using only basic notions so that it can be used both by students and researchers interested in the subject.

Category theory is the natural language that is being used to develop the theory of groupoids. However, categorical presentations of mathematical subjects tend to become highly abstract very fast and out of reach of many potential users. To avoid this, foundations of the theory, starting with simple examples, have been developed and used to study the structure of finite groups and groupoids. The appropriate language and notions from category theory have been developed for students of mathematics and theoretical physics. The book presents the theory on the same level as the ordinary and elementary theories of finite groups and their representations, and provides a unified picture of the same. The structure of the algebra of finite groupoids is analysed, along with the classical theory of characters of their representations.

Unnecessary complications in the formal presentation of the subject are avoided. The book offers an introduction to the language of category theory in the concrete setting of finite sets. It also shows how this perspective provides a common ground for various problems and applications, ranging from combinatorics, the topology of graphs, structure of databases and quantum physics.

Table of Contents


1. Categories: basic notions and examples
    Introducing the main characters
    Categories: formal definitions
    A categorical definition of groupoids and groups
    Historical notes and additional comments

2. Groups
    Groups, subgroups and normal subgroups: basic notions
    The symmetric group
    Group homomorphisms and Cayley's theorem   
    The alternating group
    Products of groups
    Historical notes and additional comments

3. Groupoids
    Groupoids: basic concepts
    Puzzles and groupoids

4. Actions of groups and groupoids
    Symmetries, groups and groupoids
    The action groupoid
    Symmetries and groupoids 
    Weinstein's tilings
    Cayley's theorem for groupoids

5. Functors and transformations
    An interlude: categories and databases
    Homomorphisms of groupoids
    Equivalence: Natural transformations

6. The structure of groupoids
    Normal subgroupoids
    Simple groupoids 
    The structure of groupoids: second structure theorem
    Classification of groupoids up to order 20
    Groupoids with Abelian isotropy group


7. Linear representations of groups
    Linear and unitary representations of groups
    Irreducible representations
    Unitary representations of groups
    Schur's lemmas for groups

8. Characters
    Orthogonality relations
    Orthogonality relations of characters
    Inequivalent representations and irreducibility criteria
    Decomposition of the regular representation
    Tensor products of representations of groups
    Tables of characters
    Canonical decomposition
    An application in quantum mechanics: spectrum degeneracy

9. Linear representations of categories
    Linear representations of categories
    Properties of representations of categories
    Linear representations of groupoids

10. Algebras and groupoids
    The algebra of a category
    The algebra of a groupoid
    Representations of Algebras
    Representations of groupoids and modules

11. Semi-simplicity
    Irreducible representations of algebras
    Semi-simple modules
    The Jordan-Holder theorem
    Semi-simple algebras: the Jacobson radical
    Characterizations of semi-simplicity
    The algebra of a finite groupoid is semi-simple

12. Representations of groupoids
    Characters again
    Operations with groupoids and representations
    The left and right regular representations of a finite groupoid
    Some simple examples


A Glossary of Linear Algebra

B Generators and relations

C Schwinger Algebra



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Alberto Ibort is full professor of Applied Mathematics in the Department of Mathematics of the Universidad Carlos III of Madrid, Spain and member of the Mathematical Institute, ICMAT, Madrid, Spain. He has been visiting professor and Fulbright Scholar at the University of California at Berkeley, USA, postdoc at the Université de Paris VI, France and the Niels Bohr Institute, Denmark, and professor of Theoretical Physics at the Universidad Complutense of Madrid. His research includes several areas of Mathematics and Mathematical Physics: Functional Analysis, Differential Geometry and more recently algebraic structures on Physics and Engineering, mainly control theory.

Miguel A. Rodríguez is full professor in the Department of Theoretical Physics of Universidad Complutense of Madrid, Spain. His teaching is mainly related to courses on Mathematics applied to Physics, in particular group theory. He has been visiting professor at Université de Montréal, Canada, University of California at Los Angeles, USA, and Università di Roma Tre, Italy. His research field includes several areas of Mathematical Physics: Integrable Systems, Group Theory, and Difference Equations.