Elements of the Representation Theory of Associative Algebras

by Daniel Simson , Andrzej Skowronski

• ISBN13: 9780521708760
• ISBN10: 0521708761
• Edition: 1st
• Format: Paperback
• Copyright: 2007-11-12
• Publisher: Cambridge University Press

Summary

The final part 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 and homological algebra. This volume provides an introduction to the representation theory of representation-infinite tilted algebras from the point of view of the time-wild dichotomy. Also included is a collection of selected results relating to the material discussed in all three volumes. The book is primarily addressed to a graduate student starting research in the representation theory of algebras, but will also be of interest to mathematicians in other fields. Proofs are presented in complete detail, and the text includes many illustrative examples and a large number of exercises at the end of each chapter, making the book suitable for courses, seminars, and self-study.

Table of Contents

Introduction	p. ix
Tubular extensions and tubular coextensions of algebras	p. 1
One-point extensions and one-point coextensions of algebras	p. 2
Tubular extensions and tubular coextensions of algebras	p. 19
Branch extensions and branch coextensions of algebras	p. 39
Tubular extensions and tubular coextensions of concealed algebras of Euclidean type	p. 45
Exercises	p. 62
Branch algebras	p. 65
Branches and finite line extensions	p. 66
Tilted algebras of an equioriented type A[subscript m]	p. 78
Exercises	p. 100
Tilted algebras of Euclidean type	p. 103
Stone cones in hereditary standard stable tubes	p. 104
Tilting with hereditary standard stable tubes	p. 108
Representation-infinite tilted algebras of Euclidean type	p. 131
Domestic tubular extensions and domestic tubular coextensions of concealed algebras of Euclidean type	p. 163
A classification of tilted algebras of Euclidean type	p. 184
A controlled property of the Euler form of tilted algebras of Euclidean type	p. 196
Exercises	p. 209
Wild hereditary algebras and tilted algebras of wild type	p. 213
Regular components	p. 215
Homomorphisms between regular modules	p. 231
Perpendicular categories	p. 249
Wild behaviour of the module category	p. 269
Tilted algebras of wild type	p. 279
Exercises	p. 305
Tame and wild representation type of algebras	p. 309
Wild representation type	p. 310
Indecomposable modules over the polynomial algebra K[t]	p. 332
Tame representation type	p. 337
Exercises	p. 356
Perspectives	p. 359
Components of the Auslander-Reiten quiver of an algebra	p. 360
The Tits quadratic form of an algebra	p. 366
Tilted and quasitilted algebras	p. 374
Algebras of small homological dimensions	p. 386
Selfinjective algebras of tilted and quasitilted type	p. 396
Related topics and research directions	p. 403
Bibliography	p. 405
Index	p. 451
List of symbols	p. 455
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