An Introduction to Groups, Groupoids and Their Representations

Author:   Alberto Ibort ,  Miguel A. Rodriguez
Publisher:   Taylor & Francis Ltd
ISBN:  

9781138035867


Pages:   18
Publication Date:   12 November 2019
Format:   Hardback
Availability:   In Print   Availability explained
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An Introduction to Groups, Groupoids and Their Representations


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Overview

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.

Full Product Details

Author:   Alberto Ibort ,  Miguel A. Rodriguez
Publisher:   Taylor & Francis Ltd
Imprint:   CRC Press
Weight:   0.726kg
ISBN:  

9781138035867


ISBN 10:   1138035866
Pages:   18
Publication Date:   12 November 2019
Audience:   Professional and scholarly ,  College/higher education ,  Professional & Vocational ,  Tertiary & Higher Education
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

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-H older 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

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Author Information

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 Universite 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. Rodriguez 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 Universite de Montreal, Canada, University of California at Los Angeles, USA, and Universita di Roma Tre, Italy. His research field includes several areas of Mathematical Physics: Integrable Systems, Group Theory, and Difference Equations.

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