Representation-finite selfinjective algebras of class An
Representation-finite selfinjective algebras of class An
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An 类表示有限自射代数
DOI：
10.1007/bfb0088479
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发表时间：
1980
影响因子：
1.8
通讯作者：
Christine Riedtmann
中科院分区：
文献类型：
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作者：
Christine Riedtmann
本文k表示代数闭域，A表示单位元为IA=I的有限维结合k-代数，mod A表示有限k维酉右A-模的CARE@CREY。我们假设A是连通的(即任何直接因子-代数不是0就是A)且表示有限。最后一个条件意味着存在有限多个非同构不可分解模MI……M r Qmod A使得每个N Emod A同构于m i mr M 1~…@M r对某些确定的m i……M E~事实上我们选择了这样的模块MI.....我一劳永逸。从第2段开始，我们进一步假设A是自射模(即A是自身上的内射右模)。我们的目的是证明关于表示有限自Jeorive代数的一些一般命题，并完全分类树类An([8]，1.7和§2，Hauptsatz]的那些命题。0，E6，E7或E8类的表示有限自紧代数的分类将在以后的文献中出现n。将树类A的子类划分为两个子类，即文[5]中已考虑的环状代数和所谓的Moebius代数。我们结果的最初推动力是观察到[5]中使用的方法可能是
Throuzhout this article k denotes an alzebraically closed field, A a finite-dimensional associative k-algebra with unit element IA= I and mod A the care@ cry of unitary right A-modules with finite k-dimension. We assume A to be connected(ie any direct factor-algebra is either 0 or A) and representation-finite. The last condition means that there are finitely many non-isomorphic indeeomposable modules MI..... M r Qmod A such that each N Emod A is isomorphic to m I mr M 1~...@ M r for some well determined m I..... m E~ In fact we r choose such modules MI..... M once for all. From paragraph 2 onwards we shall moreover assume that A is selfinjective (ie is an injective right module over itself). Our purpose is to prove some general statements on representation-finite selfinjeorive algebras and to classify completely those of tree-class A n ([8], 1.7 and § 2, Hauptsatz]. The classification of the representation-finite selfinjeetive algebras of class 0, E6, E 7 or E 8 will appear n in subsequent publioabions. Those of tree-class A are divided into o two subclasses, the wreath-like algebras already considered in [5] and the so-called Moebius algebras. The starting impetus for our results was the observation that the methods used in [5] may be