Triangulated Categories in the Representation of Finite Dimensional Algebras (London Mathematical Society Lecture Note Series, Series Number 119)

Happel presents an introduction to the use of triangulated categories in the study of representations of finit-dimensional algeras. In recent years representation theory has been an area of intense research and the author shows that derived categories of finite=dimensional algebras are a useful tool in studying tilting processes. Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and iterated tilted algebras. The author shows how triangulated categories arise naturally in the study of Frobenius categories. The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category. With a comprehensive reference section, algebraists and research students in this field will find this an indispensable account of the theory of finite-dimensional algebras. Cover......Page 1 Title......Page 4 Copyright......Page 5 TABLE OF CONTENTS......Page 6 Preface......Page 8 1. Foundations......Page 12 2. Frobenius categories......Page 21 3. Examples......Page 35 4. Auslander-Reiten triangles......Page 42 5. Description of some derived categories......Page 54 1. t-categories......Page 68 2. Repetitive algebras......Page 70 3. Generating subcategories......Page 81 4. The main theorem......Page 85 5. Examples......Page 100 CHAPTER III: Tilting theory......Page 104 1. Grothendieck groups of triangulated categories......Page 106 2. The invariance property......Page 114 3. The Brenner-Butler Theorem......Page 124 4. Torsion theories......Page 129 5. Tilted algebras......Page 144 6. Partial tilting modules......Page 152 7. Concealed algebras......Page 160 1. Piecewise hereditary algebras......Page 163 2. Cycles in mod k13......Page 171 3. The representation-finite case......Page 175 4. Iterated tilted algebras......Page 182 5. The general case......Page 186 6. The Dynkin case......Page 191 7. The affine case......Page 199 1. Preliminaries......Page 207 2. The representation-finite case......Page 210 3. The representation-infinite case......Page 212 References......Page 214 Index......Page 218 This book is an introduction to the use of triangulated categories in the study of representations of finite-dimensional algebras. In recent years representation theory has been an area of intense research and the author shows that derived categories of finite-dimensional algebras are a useful tool in studying tilting processes. Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and interated tilted algebras. The author shows how triangulated categories arise naturally in the study of Frobenius categories. The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category. With a comprehensive reference section, algebraists and research students in this field will find this an indispensable account of the theory of finite-dimensional algebras.