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GRADUATE STUDIES IN MATHEMATICS 174 Quiver Representations and Quiver Varieties Alexander Kirillov Jr. American Mathematical Society https://doi.org/10.1090//gsm/174 Quiver Representations and Quiver Varieties GRADUATE STUDIES IN MATHEMATICS 174 Quiver Representations and Quiver Varieties Alexander Kirillov Jr. American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 16G20; Secondary 14C05, 14D21, 16G60, 16G70, 17B10, 17B22, 17B67. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-174 Library of Congress Cataloging-in-Publication Data Names: Kirillov, Alexander A., 1967- Title: Quiver representations and quiver varieties / Alexander Kirillov, Jr. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Grad- uate studies in mathematics : volume 174 | Includes bibliographical references and index. Identifiers: LCCN 2016018803 | ISBN 9781470423070 (alk. paper) Subjects: LCSH: Directed graphs. | Representations of graphs. | Graph theory. | AMS: Associative rings and algebras – Representation theory of rings and algebras – Representations of quivers and partially ordered sets. msc | Algebraic geometry – Cycles and subschemes – Parametrization (Chow and Hilbert schemes). msc | Algebraic geometry – Families, fibrations – Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory). msc | Associative rings and algebras – Representation theory of rings and algebras – Representation type (finite, tame, wild, etc.). msc | Associative rings and algebras – Representation theory of rings and algebras – Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Representations, algebraic theory (weights). msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Root systems. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. msc Classification: LCC QA166.15 .K75 2016 | DDC 512/.46–dc23 LC record available at https:// lccn.loc.gov/2016018803 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 To my children: Vanya, Elena, and Andrew Contents Preface xi Part 1. Dynkin Quivers Chapter 1. Basic Theory 3 §1.1. Basic definitions 3 §1.2. Path algebra; simple and indecomposable representations 7 §1.3. K-group and dimension 11 §1.4. Projective modules and the standard resolution 11 §1.5. Euler form 15 §1.6. Dynkin and Euclidean graphs 16 §1.7. Root lattice and Weyl group 20 Chapter 2. Geometry of Orbits 23 §2.1. Representation space 23 §2.2. Properties of orbits 24 §2.3. Closed orbits 26 Chapter 3. Gabriel’s Theorem 31 §3.1. Quivers of finite type 31 §3.2. Reflection functors 32 §3.3. Dynkin quivers 38 §3.4. Coxeter element 41 §3.5. Longest element and ordering of positive roots 43 vii viii Contents Chapter 4. Hall Algebras 47 §4.1. Definition of Hall algebra 47 §4.2. Serre relations and Ringel’s theorem 52 §4.3. PBW basis 56 §4.4. Hall algebra of constructible functions 61 §4.5. Finite fields vs. complex numbers 66 Chapter 5. Double Quivers 69 §5.1. The double quiver 69 §5.2. Preprojective algebra 70 §5.3. Varieties Λ(v)72 §5.4. Composition algebra of the double quiver 75 Part 2. Quivers of Infinite Type Chapter 6. Coxeter Functor and Preprojective Representations 83 §6.1. Coxeter functor 84 §6.2. Preprojective and preinjective representations 86 §6.3. Auslander–Reiten quiver: Combinatorics 88 §6.4. Auslander–Reiten quiver: Representation theory 92 §6.5. Preprojective algebra and Auslander–Reiten quiver 96 Chapter 7. Tame and Wild Quivers 103 §7.1. Tame-wild dichotomy 103 §7.2. Representations of the cyclic quiver 105 §7.3. Affine root systems 106 §7.4. Affine Coxeter element 107 §7.5. Preprojective, preinjective, and regular representations 112 §7.6. Category of regular representations 113 §7.7. Representations of the Kronecker quiver 118 §7.8. Classification of regular representations 121 §7.9. Euclidean quivers are tame 126 §7.10. Non-Euclidean quivers are wild 127 §7.11. Kac’s theorem 129 Contents ix Chapter 8. McKay Correspondence and Representations of Euclidean Quivers 133 §8.1. Finite subgroups in SU(2) and regular polyhedra 133 §8.2. ADE classification of finite subgroups 135 §8.3. McKay correspondence 141 §8.4. Geometric construction of representations of Euclidean quivers 146 Part 3. Quiver Varieties Chapter 9. Hamiltonian Reduction and Geometric Invariant Theory 159 §9.1. Quotient spaces in differential geometry 159 §9.2. Overview of geometric invariant theory 160 §9.3. Relative invariants 163 §9.4. Regular points and resolution of singularities 168 §9.5. Basic definitions of symplectic geometry 171 §9.6. Hamiltonian actions and moment map 174 §9.7. Hamiltonian reduction 177 §9.8. Symplectic resolution of singularities and Springer resolution 180 §9.9. K¨ahler quotients 182 §9.10. Hyperk¨ahler quotients 186 Chapter 10. Quiver Varieties 191 §10.1. GIT quotients for quiver representations 191 §10.2. GIT moduli spaces for double quivers 195 §10.3. Framed representations 200 §10.4. Framed representations of double quivers 204 §10.5. Stability conditions 206 §10.6. Quiver varieties as symplectic resolutions 210 §10.7. Example: Type A quivers and flag varieties 212 §10.8. Hyperk¨ahler construction of quiver varieties 216 §10.9. C× action and exceptional fiber 219 x Contents Chapter 11. Jordan Quiver and Hilbert Schemes 225 §11.1. Hilbert schemes 225 §11.2. Quiver varieties for the Jordan quiver 227 §11.3. Moduli space of torsion free sheaves 230 §11.4. Anti-self-dual connections 235 §11.5. Instantons on R4 and ADHM construction 238 Chapter 12. Kleinian Singularities and Geometric McKay Correspondence 241 §12.1. Kleinian singularities 241 §12.2. Resolution of Kleinian singularities via Hilbert schemes 243 §12.3. Quiver varieties as resolutions of Kleinian singularities 245 §12.4. Exceptional fiber and geometric McKay correspondence 248 §12.5. Instantons on ALE spaces 253 Chapter 13. Geometric Realization of Kac–Moody Lie Algebras 259 §13.1. Borel–Moore homology 259 §13.2. Convolution algebras 261 §13.3. Steinberg varieties 264 §13.4. Geometric realization of Kac–Moody Lie algebras 266 Appendix A. Kac–Moody Algebras and Weyl Groups 273 §A.1. Cartan matrices and root lattices 273 §A.2. Weight lattice 274 §A.3. Bilinear form and classification of Cartan matrices 275 §A.4. Weyl group 276 §A.5. Kac–Moody algebra 277 §A.6. Root system 278 §A.7. Reduced expressions 280 §A.8. Universal enveloping algebra 281 §A.9. Representations of Kac–Moody algebras 282 Bibliography 285 Index 293 Preface This book is an introduction to the theory of quiver representations and quiver varieties. It is based on a course given by the author at Stony Brook University. It begins with basic definitions and ends with Nakajima’s work on quiver varieties and the geometric realization of Kac–Moody Lie algebras. The book aims to be a readable introduction rather than a monograph. Thus, while the first chapters of the book are mostly self-contained, in the second half of the book some of the more technical proofs are omitted; we only give the statements and some ideas of the proofs, referring the reader to the original papers for details. We tried to make this exposition accessible to graduate students, requir- ing only a basic knowledge of algebraic geometry, differential geometry, and the theory of Lie groups and Lie algebras. Some sections use the language of derived categories; however, we tried to reduce their use to a minimum. The material presented in the book is taken from a number of papers and books (some small parts are new). We provide references to the orig- inal works; however, we made no attempt to discuss the history of the work. In many cases the references given are the most convenient or easy to read sources, rather than the papers in which the result was first intro- duced. In particular, we heavily used Crawley-Boevey’s lectures [CB1992], Ginzburg’s notes [Gin2012], and Nakajima’s book [Nak1999]. Acknowledgments. The author would like to thank Pavel Etingof, Victor Ginzburg, Radu Laza, Hiraku Nakajima, Olivier Schiffmann, Jason Starr, and Jaimie Thind for many discussions and explanations. Without them, this book would never have been written. xi xii Preface In addition, I would also like to thank Ljudmila Kamenova and the anonymous reviewers for their comments on the preliminary version of this book and my son Andrew Kirillov for his help with proofreading. Bibliography [AM1978] R. Abraham and J. E.
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  • Persistence Modules Vs. Quiver Representations

    Persistence Modules Vs. Quiver Representations

    Persistence Modules vs. Quiver Representations k: field of coefficients 1 1 1 0 ( 0 ) ( 0 1 ) ( 1 ) ( 1 1 ) persistence module: k / k2 / k / k2 / k2 underlying graph: • a • b • c • d • 1 / 2 / 3 / 4 / 5 1 Persistence Modules vs. Quiver Representations k: field of coefficients 1 1 1 0 ( 0 ) ( 0 1 ) ( 1 ) ( 1 1 ) quiver representation: k / k2 / k / k2 / k2 quiver: • a • b • c • d • 1 / 2 / 3 / 4 / 5 1 Outline • quivers and representations • the category of representations • the classification problem • Gabriel's theorem(s) • proof of Gabriel's theorem • beyond Gabriel's theorem 2 Quivers and Representations Definition: A quiver Q consists of two sets Q0;Q1 and two maps s; t : Q1 ! Q0. The elements in Q0 are called the vertices of Q, while those roughly speaking, a quiver is a (potentially infinite) directed multigraph of Q1 are called the arrows. The source map s assigns a source sa to every arrow a 2 Q1, while the target map t assigns a target ta. Ln(n ≥ 1) • • ··· • • 1 / 2 / n/ −1 / n 3 Quivers and Representations Definition: A quiver Q consists of two sets Q0;Q1 and two maps s; t : Q1 ! Q0. The elements in Q0 are called the vertices of Q, while those roughly speaking, a quiver is a (potentially infinite) directed multigraph of Q1 are called the arrows. The source map s assigns a source sa to every arrow a 2 Q1, while the target map t assigns a target ta. •1 • ? _ a _ e b d - • • • • 2 c / 3 Q Q¯ 3 Quivers and Representations Definition: A representation of Q over a field k is a pair V = (Vi; va) consisting of a set of k-vector spaces fVi j i 2 Q0g together with a set of k-linear maps fva : Vsa ! Vta j a 2 Q1g.