Covering spaces in representation-theory
Covering spaces in representation-theory
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DOI：
10.1007/bf01396624
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发表时间：
1982-10
影响因子：
3.1
通讯作者：
K. Bongartz;P. Gabriel
中科院分区：
文献类型：
--
作者：
K. Bongartz;P. Gabriel
在1979年，Chr. Riedtmann引入了表示有限代数A的Auslander-Reiten代数FA的覆盖（[15];参见下面的1.3和2.2）。我们的主要结果是，FA在一般情况下允许许多有限覆盖，并且每一个覆盖都是某个表示有限M（2.9）的Auslander-Reiten复盖FM。为了证明第一个陈述，我们在w中证明了FA的有限覆盖是由FA的基本群H（1.2）在有限集上的作用来分类的;一般来说，有很多这样的作用，因为H是一个自由（非交换）群（4.2）。我们通过考虑由FA的有限覆盖A的网格关系定义的代数E（[16]，1.4;见下面的2.5）得到我们的第二个主要陈述;这样的E满足表征End（@）形式的代数的Auslander条件。M/），其中M i的范围是通过在某个表示有限代数上的不可分解模的所选择的表示（2.3）。当A= Fa时，本文研究了E与A之间的关系。本文所发展的理论概念产生了具体的算法（和计算机程序），使我们能够构造许多代数的Auslander-Reiten箭图。我们在W中开始这些算法，处理H= 1的特殊情况。一般情况下将审查在随后的出版物中，我们借用Auslander-Reiten箭图的14个”极大”代数列出在我们的文件的最后（每个基本连接表示有限代数与两个简单的模块是同构的商的”极大”代数或其相反）。这些极大代数的列表也是由AV Nikulin和CA Panasiuk [14]作为在基辅开发的方法的应用而获得的。本文与Chr. Riedtmann（[15]，[16]）的结果密切相关。她未出版的Auslander-Reiten箭袋集是一个决定性的帮助，证明了基本组是免费的。不幸的是，她自己在D类自内射代数方面的工作，以及波士顿和齐里希之间的距离，最终阻止了我们进行共同出版的最初计划。我们高兴地感谢她的赞扬和评论。
In 1979 Chr. Riedtmann introduced coverings of the Auslander-Reiten quiver FA of a representation-finite algebra A ([15]; see also 1.3 and 2.2 below). Our main results are that FA admits many finite coverings in general, and that each of these is the Auslander-Reiten quiver FM of some representation-finite M (2.9). In order to prove the first statement we show in w that the finite coverings of FA are classified by the actions of the fundamental group H (1.2) of F a on finite sets; in general, there are many such actions because H is a free (non-commutative) group (4.2). We obtain our second main statement by considering the algebra E which is defined by the mesh relations of a finite covering A of FA ([16], 1.4; see 2.5 below); such an E satisfies the Auslander conditions characterizing the algebras of the form End (@. M/), where M i ranges through chosen representatives of the indecomposable modules over some representation-finite algebra (2.3). In case A= Fa, the relations between E and A are studied in w 5.The theoretical notions developed in this paper give rise to concrete algorithms (and computer programs) which enable us to construct the Auslander-Reiten quivers for plenty of algebras. We enter upon these algorithms in w tackling the special case H= 1. The general case will be examined in a subsequent publication, from which we borrow the Auslander-Reiten quivers of the 14" maximal" algebras listed at the end of our paper (each basic connected representation-finite algebra with two simple modules is isomorphic to a quotient of a" maximal" algebra or to its opposite). The list of these maximal algebras has also been obtained by AV Nikulin and CA Panasiuk [14] as an application of the methods developed in Kiev. The present paper is intimately related to the results of Chr. Riedtmann ([15],[16]). Her unpublished collection of Auslander-Reiten quivers was a decisive help in proving that the fundamental group is free. Unfortunately, her own work on selfinjective algebras of class D, and the distance between Boston and Ziirich finally prevented us from carrying through the original plan of a common publication. We take pleasure in thanking her for encouragements and remarks.