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Basic Representation Theory of Algebras

  • Textbook
  • © 2020

Overview

Authors:
  1. Ibrahim Assem

    1. Département de mathématiques, Université de Sherbrooke, Sherbrooke, Canada

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    You can also search for this author in PubMed   Google Scholar

  2. Flávio U. Coelho

    1. IME, University of Sao Paulo, São Paulo, Brazil

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Highlights the interplay between Auslander–Reiten theory and the radical of the module category
  • Offers detailed examples and concrete computations that support its approach
  • Can be covered in a one-semester course while also being comprehensive

Part of the book series: Graduate Texts in Mathematics (GTM, volume 283)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Modules, algebras and quivers

    • Ibrahim Assem, Flávio U. Coelho
    Pages 1-40

  3. The radical and almost split sequences

    • Ibrahim Assem, Flávio U. Coelho
    Pages 41-98

  4. Constructing almost split sequences

    • Ibrahim Assem, Flávio U. Coelho
    Pages 99-155

  5. The Auslander–Reiten quiver of an algebra

    • Ibrahim Assem, Flávio U. Coelho
    Pages 157-234

  6. Endomorphism algebras

    • Ibrahim Assem, Flávio U. Coelho
    Pages 235-269

  7. Representation-finite algebras

    • Ibrahim Assem, Flávio U. Coelho
    Pages 271-304

  8. Back Matter

    Pages 305-311
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Keywords

About this book

This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander–Reiten theory and the study of the radical of a module category. It starts by introducing and describing several characterisations of the radical of a module category, then presents the central concepts of irreducible morphisms and almost split sequences, before providing the definition of the Auslander–Reiten quiver, which encodes much of the information on the module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory. The book ends with selected properties of representation-finite algebras, which are now the best understood class of algebras. 
Intended for graduate students in representation theory, this book is also of interest to any mathematician wanting to learn the fundamentals of this rapidly growing field. A graduate course innon-commutative or homological algebra, which is standard in most universities, is a prerequisite for readers of this book.





Reviews

“This text is a well-conceived and accessible entry point to the representation theory of finite-dimensional algebras, taking the modern perspective of focussing on morphisms between modules rather than just modules themselves.” (Ryan David Kinser, Mathematical Reviews, December, 2021)

Authors and Affiliations

  • Département de mathématiques, Université de Sherbrooke, Sherbrooke, Canada

    Ibrahim Assem

  • IME, University of Sao Paulo, São Paulo, Brazil

    Flávio U. Coelho

About the authors

Ibrahim Assem obtained his PhD. from Carleton University, Canada, in 1981, and he has taught mathematics at the Université de Sherbrooke, Canada, since 1988. His main research interests are the representation theory of algebras, cluster algebras and homological algebra. He has published 115 research papers, one chapter in a collective book, four textbooks and one monograph.

Flávio Ulhoa Coelho has taught at the University of São Paulo, Brazil, since 1985. He obtained his PhD. from the University of Liverpool, UK in 1990. He has been a Full Professor since 2003 and was the director of USP's Institute of Mathematics and Statistics from 2010-2014. He has published over 70 research papers and three undergraduate textbooks in mathematics, as well as nine literature books.

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