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Representations of Algebras

Tame and Wild Behavior

  • Book
  • © 2022

Overview

Authors:
  1. José-Antonio de la Peña

    1. Instituto de Matemáticas, National Autonomous University of Mexico, México, Mexico

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  • Provides a self-contained introduction to a variety of topics in representation theory of algebras
  • Includes proofs of Brauer–Thrall conjectures and other topics not usually treated in introductory books
  • Presents different approaches to the tame and wild determination of finite dimensional algebras

Part of the book series: Algebra and Applications (AA, volume 30)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Introduction and First Examples

    • José-Antonio de la Peña
    Pages 1-31

  3. A Categorical Approach

    • José-Antonio de la Peña
    Pages 33-53

  4. Constructive Methods

    • José-Antonio de la Peña
    Pages 55-74

  5. Spectral Methods in Representation Theory

    • José-Antonio de la Peña
    Pages 75-112

  6. Group Actions on Algebras and Module Categories

    • José-Antonio de la Peña
    Pages 113-148

  7. Reflections and Weyl Groups

    • José-Antonio de la Peña
    Pages 149-169

  8. Simply Connected Algebras

    • José-Antonio de la Peña
    Pages 171-193

  9. Degenerations of Algebras

    • José-Antonio de la Peña
    Pages 195-217

  10. Further Comments

    • José-Antonio de la Peña
    Pages 219-227

  11. Back Matter

    Pages 229-233
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Keywords

About this book

This book offers an original introduction to the representation theory of algebras, suitable for beginning researchers in algebra. It includes many results and techniques not usually covered in introductory books, some of which appear here for the first time in book form.
 
The exposition employs methods from linear algebra (spectral methods and quadratic forms), as well as categorical and homological methods (module categories, Galois coverings, Hochschild cohomology) to present classical aspects of ring theory under new light. This includes topics such as rings with several objects, the Harada–Sai lemma, chain conditions, and Auslander–Reiten theory. Noteworthy and significant results covered in the book include the Brauer–Thrall conjectures, Drozd’s theorem, and criteria to distinguish tame from wild algebras.
 
This text may serve as the basis for a second graduate course in algebra or as an introduction to research in the field of representation theory of algebras. The originality of the exposition and the wealth of topics covered also make it a valuable resource for more established researchers.

Authors and Affiliations

  • Instituto de Matemáticas, National Autonomous University of Mexico, México, Mexico

    José-Antonio de la Peña

About the author

José-Antonio de la Peña, born in México in 1958, is a mathematician at the Universidad Autónoma de México. An expert in algebra, he has published around 150 papers and has received several honors, notably membership of El Colegio Nacional (México) since 2012.

Bibliographic Information

  • Book Title: Representations of Algebras

  • Book Subtitle: Tame and Wild Behavior

  • Authors: José-Antonio de la Peña

  • Series Title: Algebra and Applications

  • DOI: https://doi.org/10.1007/978-3-031-12288-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-031-12287-3Published: 23 October 2022

  • Softcover ISBN: 978-3-031-12290-3Published: 24 October 2023

  • eBook ISBN: 978-3-031-12288-0Published: 22 October 2022

  • Series ISSN: 1572-5553

  • Series E-ISSN: 2192-2950

  • Edition Number: 1

  • Number of Pages: XIV, 233

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Associative Rings and Algebras, Category Theory, Homological Algebra, Linear Algebra

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