**Graduate Studies in Mathematics**

Volume: 184;
2017;
344 pp;
Hardcover

MSC: Primary 16; 14; 13;

**Print ISBN: 978-1-4704-2556-2
Product Code: GSM/184**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4260-6
Product Code: GSM/184.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# An Introduction to Quiver Representations

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*Harm Derksen; Jerzy Weyman*

This book is an introduction to the representation theory of quivers
and finite dimensional algebras. It gives a thorough and modern
treatment of the algebraic approach based on Auslander-Reiten theory
as well as the approach based on geometric invariant theory. The
material in the opening chapters is developed starting slowly with
topics such as homological algebra, Morita equivalence, and Gabriel's
theorem. Next, the book presents Auslander-Reiten theory, including
almost split sequences and the Auslander-Reiten transform, and gives a
proof of Kac's generalization of Gabriel's theorem. Once this basic
material is established, the book goes on with developing the
geometric invariant theory of quiver representations. The book
features the exposition of the saturation theorem for semi-invariants
of quiver representations and its application to Littlewood-Richardson
coefficients. In the final chapters, the book exposes tilting modules,
exceptional sequences and a connection to cluster categories.

The book is suitable for a graduate course in quiver
representations and has numerous exercises and examples throughout the
text. The book will also be of use to experts in such areas as
representation theory, invariant theory and algebraic geometry, who
want to learn about applications of quiver representations to their
fields.

#### Readership

Graduate students and researchers interested in representation theory, quivers, and applications to categories.

#### Reviews & Endorsements

This book serves as an introductory text on quiver representations which would allow a person without any knowledge of Artin algebras to learn the subject quickly.

-- Queqing Chen, Mathematical Reviews

#### Table of Contents

# Table of Contents

## An Introduction to Quiver Representations

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter 1. Introduction 112
- Chapter 2. Homological Algebra of Quiver Representations 1930
- Chapter 3. Finite Dimensional Algebras 3546
- Chapter 4. Gabriel’s Theorem 4960
- Chapter 5. Almost Split Sequences 7384
- Chapter 6. Auslander-Reiten Theory 97108
- 6.1. Injective Envelopes and Projective Covers 97108
- 6.2. The Transpose Functor 100111
- 6.3. The Translation Functor for Quivers 102113
- 6.4. Auslander-Reiten Duality 103114
- 6.5. Coxeter Functors Revisited 107118
- 6.6. The Auslander-Reiten Quiver for Hereditary Algebras 111122
- 6.7. The Preprojective Algebra 114125
- 6.8. Bibliographical Remarks 116127

- Chapter 7. Extended Dynkin Quivers 117128
- Chapter 8. Kac’s Theorem 131142
- Chapter 9. Geometric Invariant Theory 149160
- 9.1. Algebraic Group Actions 150161
- 9.2. Linearly Reductive Groups 155166
- 9.3. The Geometry of Quotients 162173
- 9.4. Semi-Invariants and the Sato-Kimura Lemma 164175
- 9.5. Geometric Invariant Theory 167178
- 9.6. The Hilbert-Mumford Criterion 169180
- 9.7. GIT for Quiver Representations 172183
- 9.8. GIT Quotients with Respect to Weights 176187
- 9.9. Bibliographical Remarks 182193

- Chapter 10. Semi-invariants of Quiver Representations 183194
- 10.1. Background from Classical Invariant Theory 184195
- 10.2. The Le Bruyn-Procesi Theorem 187198
- 10.3. Background from the Representation Theory of \GL_{𝑛} 191202
- 10.4. Semi-invariants and Representation Theory 197208
- 10.5. Examples for Dynkin Quivers 199210
- 10.6. Schofield Semi-invariants 204215
- 10.7. The Main Theorem and Saturation Theorem 206217
- 10.8. Proof of the Main Theorem 211222
- 10.9. Semi-invariants for Dynkin Quivers 216227
- 10.10. Semi-invariants for Extended Dynkin Types 218229
- 10.11. More Examples of Rings of Semi-invariants 225236
- 10.12. Schofield Incidence Varieties 231242
- 10.13. Bibliographical Remarks 240251

- Chapter 11. Orthogonal Categories and Exceptional Sequences 243254
- 11.1. Schur Representations 244255
- 11.2. The Canonical Decomposition 246257
- 11.3. Tilting Modules 254265
- 11.4. Orthogonal Categories 259270
- 11.5. Quivers with Two Vertices 266277
- 11.6. Two Sincerity Results 269280
- 11.7. The Braid Group Action on Exceptional Sequences 270281
- 11.8. Examples 273284
- 11.9. An Algorithm for the Canonical Decomposition 275286
- 11.10. Bibliographical Remarks 285296

- Chapter 12. Cluster Categories 287298
- 12.1. A Combinatorial Model for Type 𝐴_{𝑛} 288299
- 12.2. Cluster Combinatorics and Decorated Representations 294305
- 12.3. Triangulated Categories and Derived Categories 303314
- 12.4. The Derived Category of Quiver Representations 310321
- 12.5. Cluster Categories 316327
- 12.6. Cluster Tilted Algebras 318329
- 12.7. Bibliographical Remarks 322333

- Notation 325336
- Index 327338
- Bibliography 331342
- Back Cover Back Cover1346