Elements of the Representation Theory of

This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

0. Introduction

vii

I. Algebras and modules

1

(40)

I.1. Algebras

1

(5)

I.2. Modules

6

(7)

I.3. Semisimple modules and the radical of a module

13

(5)

I.4. Direct sum decompositions

18

(7)

I.5. Projective and injective modules

25

(7)

I.6. Basic algebras and embeddings of module categories

32

(6)

I.7. Exercises

38

(3)

II. Quivers and algebras

41

(28)

II.1. Quivers and path algebras

41

(12)

II.2. Admissible ideals and quotients of the path algebra

53

(6)

II.3. The quiver of a finite dimensional algebra

59

(6)

II.4. Exercises

65

(4)

III. Representations and modules

69

(28)

III.1. Representations of bound quivers

69

(7)

III.2. The simple, projective, and injective modules

76

(10)

III.3. The dimension vector of a module and the Euler characteristic

86

(7)

III.4. Exercises

93

(4)

IV. Auslander-Reiten theory

97

(87)

IV.1. Irreducible morphisms and almost split sequences

98

(9)

IV.2. The Auslander-Reiten translations

107

(13)

IV.3. The existence of almost split sequences

120

(5)

IV.4. The Auslander-Reiten quiver of an algebra

125

(13)

IV.5. The first Brauer-Thrall conjecture

138

(5)

IV.6. Functorial approach to almost split sequences

143

(11)

IV.7. Exercises

154

(6)

V. Nakayama algebras and representation-finite group algebras

160

(24)

V.1. The Loewy series and the Loewy length of a module

160

(3)

V.2. Uniserial modules and right serial algebras

163

(5)

V.3. Nakayama algebras

168

(5)

V.4. Almost split sequences for Nakayama algebras

173

(2)

V.5. Representation-finite group algebras

175

(5)

V.6. Exercises

180

(4)

VI. Tilting theory

184

(59)

VI.1. Torsion pairs

184

(8)

VI.2. Partial tilting modules and tilting modules

192

(10)

VI.3. The tilting theorem of Brenner and Butler

202

(13)

VI.4. Consequences of the tilting theorem

215

(10)

VI.5. Separating and splitting tilting modules

225

(7)

VI.6. Torsion pairs induced by tilting modules

232

(6)

VI.7. Exercises

238

(5)

VII. Representation-finite hereditary algebras

243

(58)

VII.1. Hereditary algebras

243

(9)

VII.2. The Dynkin and Euclidean graphs

252

(7)

VII.3. Integral quadratic forms

259

(6)

VII.4. The quadratic form of a quiver

265

(13)

VII.5. Reflection functors and Gabriel's theorem

278

(20)

VII.6. Exercises

298

(3)

VIII. Tilted algebras

301

(56)

VIII.1. Sections in translation quivers

301

(5)

VIII.2. Representation-infinite hereditary algebras

306

(11)

VIII.3. Tilted algebras

317

(9)

VIII.4. Projectives and injectives in the connecting component

326

(12)

VIII.5. The criterion of Liu and Skowronski

338

(12)

VIII.6. Exercises

350

(7)

IX. Directing modules and postprojective components

357

(47)

IX.1. Directing modules

357

(4)

IX.2. Sincere directing modules

361

(5)

IX.3. Representation-directed algebras

366

(6)

IX.4. The separation condition

372

(7)

IX.5. Algebras such that all projectives are postprojective

379

(10)

IX.6. Gentle algebras and tilted algebras of type An

389

(11)

IX.7. Exercises

400

(4)

A. Appendix. Categories, functors, and homology

404

(33)

A.1. Categories

404

(5)

A.2. Functors

409

(11)

A.3. The radical of a category

420

(4)

A.4. Homological algebra

424

(7)

A.5. The group of extensions

431

(4)

A.6. Exercises

435

(2)

Bibliography

437

(14)

Index

451

(6)

List of symbols

457

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Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory

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