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

    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.


    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




    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.