An Introduction to Groups, Groupoids and Their Representations

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
    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
    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-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
    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.