An Introduction to Quiver Representations
About this Title
Harm Derksen, University of Michigan, Ann Arbor, MI and Jerzy Weyman, University of Connecticut, Storrs, CT
Publication: Graduate Studies in Mathematics
Publication Year: 2017; Volume 184
ISBNs: 978-1-4704-2556-2 (print); 978-1-4704-4260-6 (online)
MathSciNet review: MR3727119
MSC: Primary 16-02; Secondary 13A50, 14L24, 16G10, 16G20, 16G70
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.
Graduate students and researchers interested in representation theory, quivers, and applications to categories.
Table of Contents
- Homological algebra of quiver representations
- Finite dimensional algebras
- Gabriel’s theorem
- Almost split sequences
- Auslander-Reiten theory
- Extended Dynkin quivers
- Kac’s theorem
- Geometric invariant theory
- Semi-invariants of quiver representations
- Orthogonal categories and exceptional sequences
- Cluster categories