結合代數表示論基礎 第1卷（簡體書）

《結合代數表示論基礎(第1卷)(英文版)》內容簡介：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《結合代數表示論基礎》是一部三卷集的研究生水平的複合代數入門書籍，是《倫敦數學學會學生教程》系列之一。本書第一卷，主要講述表示論技巧，給出了封閉域上有限維複合代數表示論的現代技巧，從箭圖和同調代數的線性表示角度講述本論題。本書自成體系，探討該科目的最基本現代應用，例如，箭圖理論技巧，覆蓋理論和積分二次型。大量的例子和每章末的練習使書中的內容更加豐富，容易理解。詳細的證明是初學者和自學者以及想更加詳細瞭解複合代數表示論知識的讀者相當十分有益。目次：代數和模型；箭圖和代數；表示論和模型；auslander-reirten理論；nakayama代數和表示-有限群代數；tilting理論；表示有限遺傳代數；覆蓋代數；直向模。

作者：（加拿大）阿瑟姆（Assem.I.）

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