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

ISBN-10: 0521586313
ISBN-13: 9780521586313
Edition: 2005
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Description: 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  More...

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Book details

List price: $62.00
Copyright year: 2005
Publisher: Cambridge University Press
Publication date: 2/13/2006
Binding: Paperback
Pages: 472
Size: 6.00" wide x 8.75" long x 1.25" tall
Weight: 1.386

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.

Daniel Simson is a Professor of mathematics at Nicolaus Copernicus University, Torun.

Introduction
Algebras and modules
Algebras
Modules
Semisimple modules and the radical of a module
Direct sum decompositions
Projective and injective modules
Basic algebras and embeddings of module categories
Exercises
Quivers and algebras
Quivers and path algebras
Admissible ideals and quotients of the path algebra
The quiver of a finite dimensional algebra
Exercises
Representations and modules
Representations of bound quivers
The simple, projective, and injective modules
The dimension vector of a module and the Euler characteristic
Exercises
Auslander-Reiten theory
Irreducible morphisms and almost split sequences
The Auslander-Reiten translations
The existence of almost split sequences
The Auslander-Reiten quiver of an algebra
The first Brauer-Thrall conjecture
Functorial approach to almost split sequences
Exercises
Nakayama algebras and representation-finite group algebras
The Loewy series and the Loewy length of a module
Uniserial modules and right serial algebras
Nakayama algebras
Almost split sequences for Nakayama algebras
Representation-finite group algebras
Exercises
Tilting theory
Torsion pairs
Partial tilting modules and tilting modules
The tilting theorem of Brenner and Butler
Consequences of the tilting theorem
Separating and splitting tilting modules
Torsion pairs induced by tilting modules
Exercises
Representation-finite hereditary algebras
Hereditary algebras
The Dynkin and Euclidean graphs
Integral quadratic forms
The quadratic form of a quiver
Reflection functors and Gabriel's theorem
Exercises
Tilted algebras
Sections in translation quivers
Representation-infinite hereditary algebras
Tilted algebras
Projectives and injectives in the connecting component
The criterion of Liu and Skowronski
Exercises
Directing modules and postprojective components
Directing modules
Sincere directing modules
Representation-directed algebras
The separation condition
Algebras such that all projectives are postprojective
Gentle algebras and tilted algebras of type A[subscript n]
Exercises
Categories, functors, and homology
Categories
Functors
The radical of a category
Homological algebra
The group of extensions
Exercises
Bibliography
Index
List of symbols

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