代数表示论简介与综述

An Introduction and Survey on Representation Theory of Algebras

代数表示论是本世纪七十年代初兴起的代数学的一个新的分支。它的基本内容是研究一个Artin代数上的模范畴。由于各国代数学家的共同努力,这一理论于最近二十年来有了异常迅猛的发展并逐步趋于完善。本文介绍了代数表示论的理论基础:几乎可裂序列;箭图和赋值箭图的表示;Coxeter函子;AR-箭图的覆盖及代数的Galois覆盖。并简单介绍了在代数表示论中普遍应用的工具:Tilt理论,以及著名的Dpozg定理证明过程中用到的Bocs理论。

Representation theory of algebras is a new branch of algebra which began at the seventies of this centry. The main contents is to study the module category over an Artin algebra. A great deal of progress has been made since 1970's. In this paper the basic theory of representation theory of algebras, almost split sequence, the representation of quivers and valued quivers, Coxeter functors, the covering of AR-quivers and the Galois covering of algebras are introduced. The Tilt theory used widely in the representation theory and the Bocs theory which has been used to prove the famous Drozg theorem are introduced.