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

by Ibrahim Assem , Andrzej Skowronski , Daniel Simson

• ISBN13: 9780521584234
• ISBN10: 052158423X
• eBook ISBN(s): 9780511343544
• Format: Hardcover
• Copyright: 2006-02-13
• Publisher: Cambridge University Press
• More Book Details

Summary

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.

Table of Contents

0. Introduction	vii
I. Algebras and modules	1	(40)
1.1. Algebras	1	(5)
1.2. Modules	6	(7)
1.3. Semisimple modules and the radical of a module	13	(5)
1.4. Direct sum decompositions	18	(7)
1.5. Projective and injective modules	25	(7)
1.6. Basic algebras and embeddings of module categories	32	(6)
1.7. 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	(63)
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