An Introduction to Quiver Representations

Description

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. Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 10 1.1. Basic Definitions and Examples......Page 12 1.2. The Category of Quiver Representations......Page 14 1.3. Representation Spaces......Page 17 1.4. Indecomposable Representations......Page 19 1.5. The Path Algebra......Page 22 1.6. Duality......Page 25 1.7. The Krull-Remak-Schmidt Theorem......Page 26 1.8. Bibliographical Remarks......Page 28 2.1. Projective and Injective Modules......Page 30 2.2. Projective and Injective Quiver Representations......Page 33 2.3. The Hereditary Property of Path Algebras......Page 35 2.4. The Extensions Group......Page 38 2.5. The Euler Form......Page 43 2.6. Bibliographical Remarks......Page 45 3.1. Quivers with Relations......Page 46 3.2. The Jacobson Radical......Page 49 3.3. Basic Algebras......Page 52 3.4. Morita Equivalence......Page 54 3.5. Bibliographical Remarks......Page 58 Chapter 4. Gabriel’s Theorem......Page 60 4.1. Quivers of Finite Representation Type......Page 61 4.2. Dynkin Graphs......Page 63 4.3. The Reflection Functors......Page 68 4.4. The Coxeter Functor......Page 75 4.5. Examples......Page 80 4.6. Bibliographical Remarks......Page 82 5.1. Ideals of Morphisms in the Module Categories......Page 84 5.2. Irreducible Morphisms......Page 88 5.3. The Auslander-Reiten Quiver......Page 94 5.4. The Notion of an Almost Split Sequence......Page 97 5.5. Bibliographical Remarks......Page 105 6.1. Injective Envelopes and Projective Covers......Page 108 6.2. The Transpose Functor......Page 111 6.3. The Translation Functor for Quivers......Page 113 6.4. Auslander-Reiten Duality......Page 114 6.5. Coxeter Functors Revisited......Page 118 6.6. The Auslander-Reiten Quiver for Hereditary Algebras......Page 122 6.7. The Preprojective Algebra......Page 125 6.8. Bibliographical Remarks......Page 127 Chapter 7. Extended Dynkin Quivers......Page 128 7.1. Representations of the Kronecker Quiver......Page 129 7.2. The Auslander-Reiten Quiver of the Kronecker Quiver......Page 132 7.3. AR Quivers for other Extended Dynkin Types......Page 133 7.4. Bibliographical Remarks......Page 140 8.1. Deformed Preprojective Algebras......Page 142 8.2. Reflections......Page 147 8.3. Root Systems......Page 149 8.4. Quiver Representations over Finite Fields......Page 153 8.5. Bibliographical Remarks......Page 158 Chapter 9. Geometric Invariant Theory......Page 160 9.1. Algebraic Group Actions......Page 161 9.2. Linearly Reductive Groups......Page 166 9.3. The Geometry of Quotients......Page 173 9.4. Semi-Invariants and the Sato-Kimura Lemma......Page 175 9.5. Geometric Invariant Theory......Page 178 9.6. The Hilbert-Mumford Criterion......Page 180 9.7. GIT for Quiver Representations......Page 183 9.8. GIT Quotients with Respect to Weights......Page 187 9.9. Bibliographical Remarks......Page 193 Chapter 10. Semi-invariants of Quiver Representations......Page 194 10.1. Background from Classical Invariant Theory......Page 195 10.2. The Le Bruyn-Procesi Theorem......Page 198 10.3. Background from the Representation Theory of \GL_{��}......Page 202 10.4. Semi-invariants and Representation Theory......Page 208 10.5. Examples for Dynkin Quivers......Page 210 10.6. Schofield Semi-invariants......Page 215 10.7. The Main Theorem and Saturation Theorem......Page 217 10.8. Proof of the Main Theorem......Page 222 10.9. Semi-invariants for Dynkin Quivers......Page 227 10.10. Semi-invariants for Extended Dynkin Types......Page 229 10.11. More Examples of Rings of Semi-invariants......Page 236 10.12. Schofield Incidence Varieties......Page 242 10.13. Bibliographical Remarks......Page 251 Chapter 11. Orthogonal Categories and Exceptional Sequences......Page 254 11.1. Schur Representations......Page 255 11.2. The Canonical Decomposition......Page 257 11.3. Tilting Modules......Page 265 11.4. Orthogonal Categories......Page 270 11.5. Quivers with Two Vertices......Page 277 11.6. Two Sincerity Results......Page 280 11.7. The Braid Group Action on Exceptional Sequences......Page 281 11.8. Examples......Page 284 11.9. An Algorithm for the Canonical Decomposition......Page 286 11.10. Bibliographical Remarks......Page 296 Chapter 12. Cluster Categories......Page 298 12.1. A Combinatorial Model for Type ��_{��}......Page 299 12.2. Cluster Combinatorics and Decorated Representations......Page 305 12.3. Triangulated Categories and Derived Categories......Page 314 12.4. The Derived Category of Quiver Representations......Page 321 12.5. Cluster Categories......Page 327 12.6. Cluster Tilted Algebras......Page 329 12.7. Bibliographical Remarks......Page 333 Notation......Page 336 Index......Page 338 Bibliography......Page 342 Back Cover......Page 346