Quiver Representations

1. Front Matter

   Pages i-xi

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2. Quivers and Their Representations

   1. Front Matter

      Pages 1-1

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   2. Representations of Quivers

      • Ralf Schiffler

      Pages 3-31

   3. Projective and Injective Representations

      • Ralf Schiffler

      Pages 33-68

   4. Examples of Auslander–Reiten Quivers

      • Ralf Schiffler

      Pages 69-105

3. Path Algebras

   1. Front Matter

      Pages 107-107

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   2. Algebras and Modules

      • Ralf Schiffler

      Pages 109-131

   3. Bound Quiver Algebras

      • Ralf Schiffler

      Pages 133-151

   4. New Algebras from Old

      • Ralf Schiffler

      Pages 153-173

   5. Auslander–Reiten Theory

      • Ralf Schiffler

      Pages 175-201

   6. Quadratic Forms and Gabriel’s Theorem

      • Ralf Schiffler

      Pages 203-222

4. Back Matter

   Pages 223-230

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This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The text has two parts. In Part I, the theory is studied in an elementary way using quivers and their representations. This is a very hands-on approach and requires only basic knowledge of linear algebra. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible. Part II then uses the language of algebras and modules to build on the material developed before. The equivalence of the two approaches is proved in the text. The last chapter gives a proof of Gabriel’s Theorem. The language of category theory is developed along the way as needed.

• Associative algebra
• Auslander-Reiten theory
• Gabriel's theorem
• Module
• Quiver representation
• combinatorics

“This book is an excellent text for undergraduates or beginning graduate students. The virtues of the book can be amplified by an instructor willing to go faster for those who have some prior exposure to basic algebra, or to go slower for students starting ab ovo. Secondly, a non-expert (in representation theory of quivers) may also benefit from this book in several ways … . a reader will enjoy the clear and concise overview preceding each chapter and section.” (Alex Martsinkovsky, Mathematical Reviews, February, 2016)

“The book under review is an elementary introduction to the diagrammatic or quiver approach to the representation theory of finite-dimensional algebras. It is perhaps the first such textbook addressed to advanced undergraduates or beginning graduate students. … Teaching a course from this book should be a pleasant experience. Sets of problems are provided at the end of every one of its chapters, and little notes point to the literature. For a motivated student, thebook is well suited for self-study.” (Felipe Zaldivar, MAA Reviews, December, 2014)

• Department of Mathematics, University of Connecticut, Storrs, USA

  Ralf Schiffler

Ralf Schiffler is a Professor in the Department of Mathematics at the University of Connecticut.

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