Investigation of algebras of infinite representation type

无限表示型代数的研究

• 批准号: 12640014
• 负责人: YAMAGATA Kunio
• 金额: $ 2.24万
• 依托单位: 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640014/
• 关键词: finite dimensional algebra; representation; repetitive algebra; Frobenius algebra; module; socle deformation; 有限次元多元還; 自己入射多元環; ガロア被覆; 安定同値; 自己同型群; 半直積; アウスランダー・ライテンクィバー; 無限表現型

We studied representations of finite dimensional algebars over a field. In particular, we concentrated on the research of selfinjective algebas (ie Frobenius algebras) based on the theory of socle deformation studied by the joint work with A. Skowronski and the head investigator. We got the following three main results.1. We studied about the open problem called 'ZA_∽ problem' which states that an algebra with AR-components of ZA_∽ type is wild. We proved a therem which implies, as corollaries, many known sufficient conditions for algebras to have AR-components of ZA_∽ type. In particular, in the case when an algebra has no identity, we found a counterexample of the problem.2. We determined a structure of the rigid automorphism group of the repetitive algebras by finite dimensional algebras.3. We studied the module categories of selfinjecitve algebras, and we proved that a selfinjective algebra A stably equivalent to an algebra B with Galois covering by a repetitive algbra has also a Galois covering by a repetitive algebra. Moreover, we determined algebras with at least three generalized standared components, and algebras of Euclidean type.

我们研究了域上有限维代数的表示。特别地，我们集中在与A. Skowronski和首席研究员共同研究的基于社会变形理论的自交代数（即Frobenius代数）的研究。我们得到了以下三个主要结果。研究了一类具有ar分量的ZA_∽型代数的开放问题“ZA_∽问题”。我们证明了一个定理，作为推论，它蕴涵了许多已知的代数具有ZA_∽型ar分量的充分条件。特别地，在代数没有恒等式的情况下，我们找到了这个问题的一个反例。用有限维代数确定了重复代数的刚性自同构群的结构。研究了自射代数的模范畴，证明了一个自射代数a稳定等价于一个被重复代数覆盖的代数B也有一个被重复代数覆盖的伽罗瓦代数。此外，我们还确定了至少具有三个广义标准分量的代数和欧几里得型代数。

项目成果

Andrzej Skowronski: "On selfinjective artin algebras having nonperiodic generalized standard Auslander-Reiten components"Colloquium Mathematicum. (印刷中).

Andrzej Skowronski：“关于具有非周期广义标准 Auslander-Reiten 分量的自注入 artin 代数”数学研讨会（正在出版）。

Yuji Yoshino: "On degenenrations of Cohen-Macaulay modules"Journal of Algebra. 248. 272-290 (2002)

Yuji Yoshino：“论 Cohen-Macaulay 模的简并”代数杂志。

B.Kulshammer: "Some inequalities between invariants of blocks"Archiv der Mathematics (Basel). 79. 81-86 (2002)

B.Kulshammer：“块不变量之间的一些不等式”Archiv der Mathematics（巴塞尔）。

Andrzej Skowronski: "On invariability of selfinjectiv algebras of tiltied type under stable equivalences"Proceedings of American Mathematical Society. in print.

Andrzej Skowronski：“论稳定等价下倾斜型自注入代数的不变性”美国数学会论文集。

Otto Kerner: "Auslander-Reiten components containing cones"Algebras and Representation Theory. 5. 369-387 (2002)

奥托·克纳（Otto Kerner）：“包含锥体的 Auslander-Reiten 分量”代数和表示论。