1st Edition

# An Introduction to Groups, Groupoids and Their Representations

By Alberto Ibort, Miguel A. Rodriguez Copyright 2020
362 Pages
by CRC Press

362 Pages
by CRC Press

362 Pages
by CRC Press

Also available as eBook on:

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.

I WORKING WITH CATEGORIES AND GROUPOIDS

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

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

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
Functors
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

II REPRESENTATIONS OF FINITE GROUPS AND GROUPOIDS

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
Characters
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
Algebras
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-Holder theorem
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
Discussion

III APPENDICES

A Glossary of Linear Algebra

B Generators and relations

C Schwinger Algebra

Bibliography

Index

### Biography

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.