结合代数表示论基础. 第1卷

0.Introduction

I.Algebras and modules

1.1.Algebras

1.2.Modules

1.3.Semisimple modules and the radical of a module . .

1.4.Direct sum decompositions

1.5.Projective and injective modules . .

1.6.Basic algebras and embeddings of module categories

1.7.Exercises

II.Quivers and algebras

II.1.Quivers and path algebras

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

II.3.The quiver of a finite dimensional algebra

II.4.Exercises

III.Representations and modules

III.1.Representations of bound quivers

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

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

III.4.Exercises

IV.Auslander-Reiten theory

IV.1.Irreducible morphisms and almost split sequences

IV.2.The Auslander-Reiten translations

IV.3.The existence of almost split sequences

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

IV.5.The first Brauer-Thrall conjecture

IV.6.Functorial approach to almost split sequences

IV：7.Exercises

V. Nakayama algebras and representation-finite groupalgebras1

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

V.2.Uniserial modules and right serial algebras

V.3.Nakayama algebras

V.4.Almost split sequences for Nakayama algebras

V.5.Representation-finite group algebras

V.6.Exercises

CONTENTS

VI.Tilting theory

VIA.Torsion pairs

VI.2.Partial tilting modules and tilting modules

VI.3.The tilting theorem of Brenner and Butler

VIA.Consequences of the tilting theorem

VI.5.Separating and splitting tilting modules

VI.6.Torsion pairs induced by tilting modules

VI.7.Exercises

VII.Representation-finite hereditary algebras

VII.1.Hereditary algebras

VII.2.The Dynkin and Euclidean graphs

VII.3.Integral quadratic forms

VII.4.The quadratic form of a quiver

VII.5.Reflection functors and Gabriels theorem

VII.6.Exercises

VIII.Tilted algebras

VIII.1.Sections in translation quivers

VIII.2.Representation-infinite hereditary algebras

VIII.3.Tilted algebras

VIII.4.Projectives and injectives in the connecting component

VIII.5.The criterion of Liu and Skowrofiski

VIII.6.Exercises

IX. Directing modules and postprojective components

IX.1.Directing modules

IX.2.Sincere directing modules

IX.3.Representation-directed algebras

IX.4.The separation condition

IX.5.Algebras such that all projectives are postprojective

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

IX.7.Exercises

A. Appendix. Categories functors and homology

A.1.Categories

A.2.Functors

A.3.The radical of a category

A.4.Homological algebra

A.5.The group of extensions

A.6.Exercises

Bibliography

Index

List of symbols

The idea of representing a complex mathematical object by a simplerone is as old as mathematics itself. It is particularly useful in classificationproblems. For instance， a single linear transformation on a finite dimen-sional vector space is very adequately characterised by its reduction to itsrational or its Jordan canonical form. It is now generally accepted that therepresentation theory of associative algebras traces its origin to Hamiltonsdescription of the complex numbers by pairs of real numbers. During the1930s， E. Noether gave to the theory its modern setting by interpreting rep-resentations as modules. That allowed the arsenal of techniques developedfor the study of semisimple algebras as well as the language and machineryof homological algebra and category theory to be applied to representationtheory. Using these， the theory grew rapidly over the past thirty years.

研究生水平的复合代数入门书籍
自成体系，例证丰富，容易理解。