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Product Code:  GSM/174 
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eBook ISBN:  9781470435028 
Product Code:  GSM/174.E 
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Hardcover ISBN:  9781470423070 
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Product Code:  GSM/174.B 
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Hardcover ISBN:  9781470423070 
Product Code:  GSM/174 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470435028 
Product Code:  GSM/174.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470423070 
eBook ISBN:  9781470435028 
Product Code:  GSM/174.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 174; 2016; 295 ppMSC: Primary 16; Secondary 14; 17
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 selfcontained 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.
ReadershipGraduate students and researchers interested in representations theory and algebraic geometry.

Table of Contents

Part 1. Dynkin quivers

Chapter 1. Basic theory

Chapter 2. Geometry of orbits

Chapter 3. Gabriel’s theorem

Chapter 4. Hall algebras

Chapter 5. Double quivers

Part 2. Quivers of infinite type

Chapter 6. Coxeter functor and preprojective representations

Chapter 7. Tame and wild quivers

Chapter 8. McKay correspondence and representations of Euclidean quivers

Part 3. Quiver varieties

Chapter 9. Hamiltonian reduction and geometric invariant theory

Chapter 10. Quiver varieties

Chapter 11. Jordan quiver and Hilbert schemes

Chapter 12. Kleinian singularities and geometric McKay correspondence

Chapter 13. Geometric realization of Kac–Moody Lie algebras

Appendix A. Kac–Moody algebras and Weyl groups


Additional Material

Reviews

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


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

Part 1. Dynkin quivers

Chapter 1. Basic theory

Chapter 2. Geometry of orbits

Chapter 3. Gabriel’s theorem

Chapter 4. Hall algebras

Chapter 5. Double quivers

Part 2. Quivers of infinite type

Chapter 6. Coxeter functor and preprojective representations

Chapter 7. Tame and wild quivers

Chapter 8. McKay correspondence and representations of Euclidean quivers

Part 3. Quiver varieties

Chapter 9. Hamiltonian reduction and geometric invariant theory

Chapter 10. Quiver varieties

Chapter 11. Jordan quiver and Hilbert schemes

Chapter 12. Kleinian singularities and geometric McKay correspondence

Chapter 13. Geometric realization of Kac–Moody Lie algebras

Appendix A. Kac–Moody algebras and Weyl groups

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