Gabriel-Roiter measure for finite dimensional algebras

有限维代数的 Gabriel-Roiter 测度

• 批准号: 127141831
• 负责人: Dr. Bo Chen
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2010-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/127141831?language=en
• 关键词: Gabriel; Roiter; measure; finite; dimensional

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.

有限维代数表示论的主要目标是确定代数的表示类型（有限/驯服/狂野）并描述其模范畴（表示、态射和范畴的同调性质）。现在的主要工具是所谓的 Auslander-Reiten 理论。这个新提案的主题是 Gabriel-Roiter 测度，Roiter 在证明处理有限类型代数的基本布劳尔-萨尔猜想 I 时使用了该测度。最近，Ringel 建议将 Gabriel-Roiter 测度发展为任何表示类型的代数表示论的基础工具，从而为 Auslander-Reiten 理论提供替代方法。然而，到目前为止，主要在有限表示类型代数情况下的 Gabriel-Roiter 测度上获得了结果。基于我在驯化遗传代数方面获得的一些初步结果，我计划将 Gabriel-Roiter 测度理论扩展到野代数，重点关注野遗传代数，例如在李理论的应用中。