Elements of the Representation Theory of Associative Algebras

by Daniel Simson , Andrzej Skowroński

• ISBN13: 9780521544207
• ISBN10: 0521544203
• Edition: 1st
• Format: Paperback
• Copyright: 2007-09-17
• Publisher: Cambridge University Press

Summary

The second 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, geometry of tubes of indecomposable modules, and homological algebra. This volume provides an up-to-date introduction to the representation theory of the representation-infinite hereditary algebras of Euclidean type, as well as to concealed algebras of Euclidean type. The book is primarily addressed to a graduate student starting research in the representation theory of algebras, but it will also be of interest to mathematicians in other fields. The text includes many illustrative examples and a large number of exercises at the end of each of the chapters. Proofs are presented in complete detail, making the book suitable for courses, seminars, and self-study.

Table of Contents

Introduction	p. ix
Tubes	p. 1
Stable tubes	p. 2
Standard stable tubes	p. 13
Generalised standard components	p. 32
Generalised standard stable tubes	p. 35
Exercises	p. 46
Module categories over concealed algebras of Euclidean type	p. 51
The Coxeter matrix and the defect of a hereditary algebra of Euclidean type	p. 52
The category of regular modules over a hereditary algebra of Euclidean type	p. 60
The category of regular modules over a concealed algebra of Euclidean type	p. 69
The category of modules over the Kronecker algebra	p. 77
A characterisation of concealed algebras of Euclidean type	p. 84
Exercises	p. 87
Regular modules and tubes over concealed algebras of Euclidean type	p. 91
Canonical algehras of Euclidean type	p. 91
Regular modules and tubes over canonical algebras of Euclidean type	p. 106
A separating family of tubes over a concealed algebra of Euclidean type	p. 124
A controlled property of the Euler form of a concealed algebra of Euclidean type	p. 133
Exercises	p. 139
Indecomposable modules and tubes over hereditary algebras of Euclidean type	p. 143
Canonically oriented Euclidean quivers, their Coxeter matrices and the defect	p. 145
Tubes and simple regular modules over hereditary algebras of Euclidean type	p. 153
Four subspace problem	p. 197
Exercises	p. 223
Minimal repreyentation-infinite algebras	p. 227
Critical integral quadratic forms	p. 228
Minimal representation-infinite algebras	p. 231
A criterion for the infinite representation type of algebras	p. 239
A classification of concealed algebras of Euclidean type	p. 244
Exercises	p. 278
Bibliography	p. 285
Index	p. 305
List of symbols	p. 307
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