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.

本课题的目的是研究有限维代数表示范畴的代数结构。(1)通过重复代数研究了带伽罗瓦拱的代数的稳定范畴，证明了自注入性质的不变性。代数具有由拟倾斜代数的重复代数覆盖的伽罗瓦。（2）我们研究了非弗罗贝尼乌斯自注入（=拟弗罗贝尼乌斯）代数，因此我们找到了这些代数的表征，并展示了具有任意大维数的非弗罗贝尼乌斯自注入代数的示例。通过这个例子，我们知道Nakayama在1939年给出的例子是维数最小的例子。 (3)当代数具有前射分量时，这是一个开放问题。我们研究了单点可拓代数问题。我们澄清了单点扩展的前射组件的存在取决于定义扩展的模块的被加数所属的 AR 组件。

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）。

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

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

Finiteness of the strong global dimension of radical square zero algebras

• DOI: 10.2478/bf02475954
• 发表时间: 2004-03
• 期刊: Central European Journal of Mathematics
• 作者: O. Kerner;A. Skowroński;K. Yamagata;D. Zacharia