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Triangulated categories in the representation theory of finite dimensional algebras / Dieter Happel.

By: Happel, Dieter, 1953-Material type: TextTextSeries: London Mathematical Society lecture note series ; 119.Publication details: Cambridge ; New York : Cambridge University Press, 1988. Description: 1 online resource (208 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781107361362; 1107361362; 9780511892271; 0511892276; 1139881760; 9781139881760; 1107366275; 9781107366275; 1107368448; 9781107368446; 1299404065; 9781299404069; 1107363810; 9781107363816; 0511629222; 9780511629228Subject(s): Triangulated categories | Representations of algebras | Modules (Algebra) | Catégories (Mathématiques) | Représentations d'algèbres | Modules (Algèbre) | MATHEMATICS -- Algebra -- Linear | Modules (Algebra) | Representations of algebras | Triangulated categories | Algebra | Darstellungstheorie | Dimension n | Kategorie Mathematik | Triangulation | Categorieën (wiskunde) | Associatieve ringen | Algebra associativa | Algebra homologica | Représentations d'algèbres | Modules (algèbre) | Catégories (mathématiques)Genre/Form: Electronic books. | Triangulierte Kategorie. | Electronic books. Additional physical formats: Print version:: Triangulated categories in the representation theory of finite dimensional algebras.DDC classification: 512/.55 LOC classification: QA169 | .H36 1988ebOther classification: 31.23 | 31.27 | SI 320 | *16Gxx | 16-02 | 16B50 | 16D50 | 16E10 | 16Exx | 16P10 | 18-02 | 18E30 | MAT 162f | MAT 180f | SK 260 | SK 320 Online resources: Click here to access online
Contents:
Cover; Title; Copyright; Contents; Preface; CHAPTER I: Triangulated categories; 1. Foundations; 2. Frobenius categories; 3. Examples; 4. Auslander-Reiten triangles; 5. Description of some derived categories; CHAPTER II: Repetitive algebras; 1. t-categories; 2. Repetitive algebras; 3. Generating subcategories; 4. The main theorem; 5. Examples; CHAPTER III: Tilting theory; 1. Grothendieck groups of triangulated categories; 2. The invariance property; 3. The Brenner-Butler Theorem; 4. Torsion theories; 5. Tilted algebras; 6. Partial tilting modules; 7. Concealed algebras
CHAPTER IV: Piecewise hereditary algebras1. Piecewise hereditary algebras; 2. Cycles in mod kZ?; 3. The representation-finite case; 4. Iterated tilted algebras; 5. The general case; 6. The Dynkin case; 7. The affine case; CHAPTER V: Trivial extension algebras; 1. Preliminaries; 2. The representation-finite case; 3. The representation-infinite case; References; Index
Summary: 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.
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Includes bibliographical references and index.

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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.

Cover; Title; Copyright; Contents; Preface; CHAPTER I: Triangulated categories; 1. Foundations; 2. Frobenius categories; 3. Examples; 4. Auslander-Reiten triangles; 5. Description of some derived categories; CHAPTER II: Repetitive algebras; 1. t-categories; 2. Repetitive algebras; 3. Generating subcategories; 4. The main theorem; 5. Examples; CHAPTER III: Tilting theory; 1. Grothendieck groups of triangulated categories; 2. The invariance property; 3. The Brenner-Butler Theorem; 4. Torsion theories; 5. Tilted algebras; 6. Partial tilting modules; 7. Concealed algebras

CHAPTER IV: Piecewise hereditary algebras1. Piecewise hereditary algebras; 2. Cycles in mod kZ?; 3. The representation-finite case; 4. Iterated tilted algebras; 5. The general case; 6. The Dynkin case; 7. The affine case; CHAPTER V: Trivial extension algebras; 1. Preliminaries; 2. The representation-finite case; 3. The representation-infinite case; References; Index

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