Elements of the Representation Theory of Associative Algebras: Volume 2, Tubes and Concealed Algebras of Euclidean type (London Mathematical Society Student Texts, Series Number 71)

The second of a three-volume set providing a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The subject is presented from the perspective of linear representations of quivers, geometry of tubes of indecomposable modules, and homological algebra. This volume provides an up-to-date introduction to the representation theory of the representation-infinite hereditary algebras of Euclidean type, as well as to concealed algebras of Euclidean type. The book is primarily addressed to a graduate student starting research in the representation theory of algebras, but it will also be of interest to mathematicians in other fields. The text includes many illustrative examples and a large number of exercises at the end of each of the chapters. Proofs are presented in complete detail, making the book suitable for courses, seminars, and self-study. Cover 1 Half-title 3 Titlle 5 Copyright 6 Dedication 7 Contents 9 Introduction 11 Chapter X Tubes 15 X.1. Stable tubes 16 X.2. Standard stable tubes 27 X.3. Generalised standard components 46 X.4. Generalised standard stable tubes 49 X.5. Exercises 60 Chapter XI Module categories over concealed algebras of Euclidean type 65 XI.1. The Coxeter matrix and the defect of a hereditary algebra of Euclidean type 66 XI.2. The category of regular modules over a hereditary algebra of Euclidean type 74 XI.3. The category of regular modules over a concealed algebra of Euclidean type 83 XI.4. The category of modules over the Kronecker algebra 91 XI.5. A characterisation of concealed algebras of Euclidean type 98 XI.6. Exercises 101 Chapter XII Regularmodules and tubes over concealed algebras of Euclidean type 105 XII.1. Canonical algebras of Euclidean type 106 XII.2. Regular modules and tubes over canonical algebras of Euclidean type 120 XII.3. A separating family of tubes over a concealed algebra of Euclidean type 138 XII.4. A controlled property of the Euler form of a concealed algebra of Euclidean type 147 XII.5. Exercises 153 Chapter XIII Indecomposable modules and tubes over hereditary algebras of Euclidean type 157 XIII.1. Canonically oriented Euclidean quivers, their Coxeter matrices and the defect 159 XIII.2. Tubes and simple regular modules over hereditary algebras of Euclidean type 167 XIII.3. Four subspace problem 211 XIII.4. Exercises 237 Chapter XIV Minimal representation-infinite algebras 241 XIV.1. Critical integral quadratic forms 242 XIV.2. Minimal representation-infinite algebras 245 XIV.3. A criterion for the infinite representation type of algebras 253 XIV.4. A classification of concealed algebras of Euclidean type 258 Bibliography 299 Index 319 List of symbols 321 Giving a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field, this title provides an introduction to the representation theory of the representation-infinite hereditary and concealed algebras of Euclidean type