由阿瑟姆编著的《结合代数表示论基础》是一部三卷集的研究生水平的复合代数入门书籍，是《伦敦数学学会学生教程》系列之一。本书第一卷，主要讲述表示论技巧，给出了封闭域上有限维复合代数表示论的现代技巧，从箭图和同调代数的线性表示角度讲述本论题。本书自成体系，探讨该科目的最基本现代应用，例如，箭图理论技巧，覆盖理论和积分二次型。大量的例子和每章末的练习使书中的内容更加丰富，容易理解。详细的证明是初学者和自学者以及想更加详细了解复合代数表示论知识的读者相当十分有益。目次：代数和模型；箭图和代数；表示论和模型；auslander-reirten理论；nakayama代数和表示-有限群代数；tilting理论；表示有限遗传代数；覆盖代数；直向模。

0.Introduction

Ⅰ.Algebras and modules

 Ⅰ.1.Algebras

 Ⅰ.2.Modules

 Ⅰ.3.Semisimple modules and the radical of a module

 Ⅰ.4.Direct sum decompositions

 Ⅰ.5.Projective and injective modules

 Ⅰ.6.Basic algebras and embeddings of module categories

 Ⅰ.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 group algebras

 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

Ⅵ.Tilting theory

 Ⅵ.1.Torsion pairs

 Ⅵ.2.Partial tilting modules and tilting modules

 Ⅵ.3.The tilting theorem of Brenner and Butler

 Ⅵ.4.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 Gabriel's 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 Skowronski

 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, funetors, 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