Gabriel-Roiter measure for finite dimensional algebras

有限维代数表示理论的主要目标是确定一个代数的表示类型（有限/驯服/野生），并描述它的模范畴（范畴的表示、态射和同调性质）。现在的主要工具是所谓的Auslander-Reiten理论。这个新提议的主题是加布里埃尔-罗伊特测度，它曾被罗伊特用于证明基本的Brauer-Thrall猜想I，处理有限型代数。最近，Ringel建议将Gabriel-Roiter测度发展成为任何表示类型代数表示理论的基础工具，从而为Auslander-Reiten理论提供了替代方法。然而，到目前为止，得到的结果主要是关于有限表示型代数的Gabriel-Roiter测度。基于我在驯化遗传代数中获得的一些初步结果，我计划将加布里埃尔-罗伊特测度理论扩展到野生代数，重点研究野生遗传代数，例如在李理论中的应用。

The main goals of representation theory of finite dimensional algebras are to determine the representation type (finite/tame/wild) of an algebra and to describe its module category (representations, morphisms, and homological properties of the category). The main tool nowadays is the so-called Auslander-Reiten theory. The subject of this new proposal is the Gabriel-Roiter measure, which had been used by Roiter in his proof of the fundamental Brauer-Thrall conjecture I, dealing with algebras of finite type. Recently, Ringel has suggested to develop the Gabriel-Roiter measure into a foundational tool for representation theory of algebras of any representation type, thus providing alternative methods to those of Auslander-Reiten theory. So far, however, results have been obtained mainly on the Gabriel-Roiter measure in the case of algebras of finite representation type. Building on some first results I have obtained for tame hereditary algebras, I am planning to extend the theory of Gabriel-Roiter measure to wild algebras, concentrating on wild hereditary algebras that are of interest for example in applications to Lie theory.