Investigation into algebraic structure of the category of representations of finite demensional algebras

有限维代数表示范畴的代数结构研究

• 批准号: 15540012
• 负责人: YAMAGATA Kunio
• 金额: $ 2.37万
• 依托单位: National University Corporation Tokyo University of Agriculture and Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540012/
• 关键词: finite dimensional algebra; self-injective algebra; Frobenius algebra; category; representation; module; クイバー

The aim of the project is to study algebraic structures of the category of representations of finite dimensional algebras.(1) We studied the stable categories of the algebras with Galois coveings by repetitive algebras, and proved the invariance of the property that a self-injecitve algebra has a Galois covering by the repetitive algebra of a quasi-tilted algebra.(2) We studied non-Frobenius self-injecitve (=quasi-Frobenius) algebras, so that we found a characterization of those algebras and showed an example of non-Frobenius self-injective algebras with arbitrarily large dimension. By this example, we know that the example given by Nakayama in 1939 is the one with the smallest dimension.(3) It is an open problem when an algebra has a preinjective component. We studied the problem for one-point extension algebras. We clarified that the existence of preinjective components of a one-point extension depends on AR-components where summands of the module defining the extension belong.

研究了有限维代数表示范畴的代数结构。(1)用重复代数研究了具有Galois凸包的代数的稳定范畴，证明了拟倾斜代数的重复代数对自内射代数有Galois覆盖的性质的不变性。(2)研究了非Frobenius自内射(=拟Frobenius)代数，得到了这类代数的一个刻画，并给出了一个具有任意大维的非Frobenius自内射代数的例子。通过这个例子，我们知道Nakayama在1939年给出的例子是维度最小的一个。(3)当一个代数有预内射分支时，这是一个公开问题。我们研究了一点扩张代数的问题。我们阐明了单点扩张的预内射分支的存在依赖于AR-分支，而AR-分支是定义该扩张的模的和所属的分支。

项目成果

Andrzej Skowronski: "On selfinjective Artin algebras having nonperiodic generalized standard Auslander-Reiten components."Colloq.Math.. 96・2. 235-244 (2003)

Andrzej Skowronski：“关于具有非周期广义标准 Auslander-Reiten 分量的自射 Artin 代数。”Colloq.Math.. 96・2 (2003)。

Otto Kerner: "Finiteness of the stromg global dimension of radical square zero algebras."Central European Journal of Mathematics. 2・1. 103-111 (2004)

奥托·克纳（Otto Kerner）：“根式零代数的强全局维数的有限性”。《中欧数学杂志》2・1（2004）。

Invariability of selfinjective algebras of quasitilted type under stable equivalences

稳定等价下准倾斜型自注入代数的不变性

• 发表时间: 2005
• 期刊: manuscripta mathematica 119・3
• 作者: O.Kerner;A.Skowronski;K.Yamagata
• 通讯作者: K.Yamagata

Positive Galois coverings of selfinjective algebras

自射代数的正伽罗瓦覆盖

• 期刊: Advances in Mathematics (印刷中)
• 作者: A.Skowronski;K.Yamagata
• 通讯作者: K.Yamagata

Invariance of selfinjective algebras of quasitilted type under stable equivalences

稳定等价下拟倾斜型自射代数的不变性

• 发表时间: 2006
• 期刊: manuscripta mathematica 119・3
• 作者: O.Kerner;A.Skowronski;K.Yamagata
• 通讯作者: K.Yamagata