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Quiver Representations and Quiver Varieties
 
Alexander Kirillov Jr. Stony Brook University, Stony Brook, NY
Quiver Representations and Quiver Varieties
Hardcover ISBN:  978-1-4704-2307-0
Product Code:  GSM/174
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-3502-8
Product Code:  GSM/174.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-2307-0
eBook: ISBN:  978-1-4704-3502-8
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
Quiver Representations and Quiver Varieties
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Quiver Representations and Quiver Varieties
Alexander Kirillov Jr. Stony Brook University, Stony Brook, NY
Hardcover ISBN:  978-1-4704-2307-0
Product Code:  GSM/174
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-3502-8
Product Code:  GSM/174.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-2307-0
eBook ISBN:  978-1-4704-3502-8
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 Details
     
     
    Graduate Studies in Mathematics
    Volume: 1742016; 295 pp
    MSC: 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 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.

  • 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1742016; 295 pp
MSC: 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 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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