**Graduate Studies in Mathematics**

Volume: 174;
2016;
295 pp;
Hardcover

MSC: Primary 16;
Secondary 14; 17

Print ISBN: 978-1-4704-2307-0

Product Code: GSM/174

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-3502-8
Product Code: GSM/174.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Quiver Representations and Quiver Varieties

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*Alexander Kirillov Jr.*

This book is an introduction to the theory of
quiver representations and quiver varieties, starting with basic
definitions and ending with Nakajima's work on quiver varieties and
the geometric realization of Kac–Moody Lie algebras.

The first part of the book is devoted to the classical theory of
quivers of finite type. Here the exposition is mostly self-contained
and all important proofs are presented in detail. The second part
contains the more recent topics of quiver theory that are related to
quivers of infinite type: Coxeter functor, tame and wild quivers,
McKay correspondence, and representations of Euclidean quivers. In the
third part, topics related to geometric aspects of quiver theory are
discussed, such as quiver varieties, Hilbert schemes, and the
geometric realization of Kac–Moody algebras. Here some of the more
technical proofs are omitted; instead only the statements and some
ideas of the proofs are given, and the reader is referred to original
papers for details.

The exposition in the book requires 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, the use of this language is reduced to a minimum.
The many examples make the book accessible to graduate students who
want to learn about quivers, their representations, and their
relations to algebraic geometry and Lie algebras.

#### Readership

Graduate students and researchers interested in representations theory and algebraic geometry.

#### Reviews & Endorsements

The book should serve as a valuable source for readers who want to understand various levels of deep connections between quiver representations, Lie theory, quantum groups, and geometric representation theory...The beautiful results discussed in the present book touch on several mathematical areas, therefore, the inclusion of background material and several examples make it convenient to learn the subject.

-- Mátyás Domokos, Mathematical Reviews

...a concise guide to representation theory of quiver representations for beginner and advanced researchers.

-- Justyna Kosakowska, Zentralblatt Math

With an adequate background in Lie theory and algebraic geometry, the book is accessible to an interested reader...it engages the reader to fill in some arguments or to look for a result in the references. As such, the book can be used for a topics course on its subjects.

-- Felipe Zaldivar, MAA Reviews

#### Table of Contents

# Table of Contents

## Quiver Representations and Quiver Varieties

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Part 1 . Dynkin Quivers 114
- Part 2 . Quivers of Infinite Type 8194
- Chapter 6. Coxeter Functor and Preprojective Representations 8396
- Chapter 7. Tame and Wild Quivers 103116
- 7.1. Tame-wild dichotomy 103116
- 7.2. Representations of the cyclic quiver 105118
- 7.3. Affine root systems 106119
- 7.4. Affine Coxeter element 107120
- 7.5. Preprojective, preinjective, and regular representations 112125
- 7.6. Category of regular representations 113126
- 7.7. Representations of the Kronecker quiver 118131
- 7.8. Classification of regular representations 121134
- 7.9. Euclidean quivers are tame 126139
- 7.10. Non-Euclidean quivers are wild 127140
- 7.11. Kac’s theorem 129142

- Chapter 8. McKay Correspondence and Representations of Euclidean Quivers 133146

- Part 3 . Quiver Varieties 157170
- Chapter 9. Hamiltonian Reduction and Geometric Invariant Theory 159172
- 9.1. Quotient spaces in differential geometry 159172
- 9.2. Overview of geometric invariant theory 160173
- 9.3. Relative invariants 163176
- 9.4. Regular points and resolution of singularities 168181
- 9.5. Basic definitions of symplectic geometry 171184
- 9.6. Hamiltonian actions and moment map 174187
- 9.7. Hamiltonian reduction 177190
- 9.8. Symplectic resolution of singularities and Springer resolution 180193
- 9.9. Kähler quotients 182195
- 9.10. Hyperkähler quotients 186199

- Chapter 10. Quiver Varieties 191204
- 10.1. GIT quotients for quiver representations 191204
- 10.2. GIT moduli spaces for double quivers 195208
- 10.3. Framed representations 200213
- 10.4. Framed representations of double quivers 204217
- 10.5. Stability conditions 206219
- 10.6. Quiver varieties as symplectic resolutions 210223
- 10.7. Example: Type 𝐴 quivers and flag varieties 212225
- 10.8. Hyperkähler construction of quiver varieties 216229
- 10.9. \Ctimes action and exceptional fiber 219232

- Chapter 11. Jordan Quiver and Hilbert Schemes 225238
- Chapter 12. Kleinian Singularities and Geometric McKay Correspondence 241254
- Chapter 13. Geometric Realization of Kac–Moody Lie Algebras 259272
- Appendix A. Kac–Moody Algebras and Weyl Groups 273286
- A.1. Cartan matrices and root lattices 273286
- A.2. Weight lattice 274287
- A.3. Bilinear form and classification of Cartan matrices 275288
- A.4. Weyl group 276289
- A.5. Kac–Moody algebra 277290
- A.6. Root system 278291
- A.7. Reduced expressions 280293
- A.8. Universal enveloping algebra 281294
- A.9. Representations of Kac–Moody algebras 282295

- Bibliography 285298
- Index 293306

- Back Cover Back Cover1311