Tame algebras with strongly simply connected Galois coverings
Tame algebras with strongly simply connected Galois coverings
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使用强简单连通的伽罗瓦覆盖来驯服代数
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发表时间：
1997
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通讯作者：
A. Skowroński
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作者：
A. Skowroński
在整个代数中，我们指的是代数闭域K上有1的基本连通有限维结合K-代数。在代数A上，我们指的是有限K维的右A-模。根据Drozd著名的Tame和Wild定理[14]，这类代数可以分为两个互不相交的类。其中一类由驯服代数组成，其不可分解模在每个维d中出现在有限个离散族和有限个单参数族中。第二类是由野代数和两个非交换自同态组成的，它们的表示理论和有限维向量空间的研究一样复杂，它们的分类是一个众所周知的未解决的问题。因此，我们可以希望只对Tame代数的模进行分类。这里，我们讨论了具有单连通Galois覆盖的Tame代数的表示理论。在Tame代数中，我们可以区分一类表示-有限代数，它只有有限个不可分解模的同构类。这类代数目前已被很好地理解(见[3]、[8]、[9]、[10])。特别地，我们知道每个表示有限代数A都有一个标准型A[10]，它是A的最好的可能退化，使得A和A有相同数目的不可分解模的同构类，并且A有一个(强)单连通Galois覆盖。这导致了Bongartz关于有限表示型的判据[8]，并将对任意表示有限代数上的模的研究归结为对相应的单连通代数的研究。驯服表示-无限代数的表示理论才刚刚出现。目前最可达的似乎是多项式增长的(Tame)代数[26]，它存在一个正整数m，使得单参数不可分解模族的个数在每一维d由d有界。它包含了国内已有的一类不可分解模类。
Throughout, by an algebra we mean a basic connected, finite-dimensional associative K-algebra with 1 over an algebraically closed field K. By a module over an algebra A we mean a right A-module of finite K-dimension. From Drozd’s remarkable Tame and Wild Theorem [14] the class of algebras may be divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory is as complicated as the study of finite-dimensional vector spaces together with two non-commuting endomorphisms, for which the classification is a well-known unsolved problem. Hence, we can hope to classify the modules only for tame algebras. Here, we are concerned with the representation theory of tame algebras having simply connected Galois coverings. Among tame algebras we may distinguish the class of representation-finite algebras, having only finitely many isomorphism classes of indecomposable modules. This class of algebras is presently rather well understood (see [3], [8], [9], [10]). In particular, we know that every representation-finite algebra A admits a standard form A [10], which is the best possible degeneration of A, such that A and A have the same number of isomorphism classes of indecomposable modules, and A admits a (strongly) simply connected Galois covering. This leads to the criterion of Bongartz for finite representation type [8], and reduces the study of modules over arbitrary representation-finite algebras to that for the corresponding simply connected algebras. The representation theory of tame representation-infinite algebras is only emerging. At present the most accessible seem to be the (tame) algebras of polynomial growth [26], for which there exists a positive integer m such that the number of one-parameter families of indecomposable modules is bounded, in each dimension d, by d. It contains the class of domestic al-