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Investigation of algebras of infinite representation type

Research Project

Project/Area Number 12640014
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTokyo University of Agriculture and Technology

Principal Investigator

YAMAGATA Kunio  Tokyo University of Agriculture and Technology, Department of Technology, Professor, 工学部, 教授 (60015849)

Co-Investigator(Kenkyū-buntansha) YOSHINO Yuji  Okayama University, Department of Mathematics, Professor, 理学部, 教授 (00135302)
MAEDA Hironobu  Tokyo University of Agriculture and Technology, Department of Technology, Associate Professor, 工学部, 助教授 (50173711)
WADA Tomoyuki  Tokyo University of Agriculture and Technology, Department of Technology, Professor, 工学部, 教授 (30134795)
KAWATA Shigeto  Osaka City University, Department of Mathematics, Associate Professor, 理学部, 助教授 (50195103)
TSUSHIMA Yukio  Osaka City University, Department of Mathematics, Professor, 理学部, 教授 (80047240)
合田 洋  東京農工大学, 工学部, 助教授 (60266913)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsfinite 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
Research Abstract

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.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (14 results)

All Other

All Publications (14 results)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kunio Yamagata: "The rigid automorphism group of a repetitive algebra"Representations of Algebras (Beijing Normal Univ.Press). II. 486-495 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Andrzej Skowronski: "On invariability of self-injective algebras of tilted type under stable equivalences"Proceedings of American Math.Soc.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Andrzej Skowronski: "On selfinjective artin algebras having nonperiodic generalized standard Auslander-Reiten components"Colloquium Mathematicum. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] B.Kulshammer: "Some inequalities between invariants of blocks"Archiv der Mathematics (Basel). 79. 81-86 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Otto Kerner: "Auslander-Reiten components containing cones"Algebras and Representatin Theory. 5. 369-387 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kunio Yamagata: "The rigid automorphism group of a repetitive algebra, Representations of Algebras II"Beijing Normal University Press. 486-495 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Andrzej Skowronski: "On invariability of selfinjectiv algebras of tiltied type under stable equivalences"Proceedings of American Mathematical Society. in print.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Andrzej Skowronski: "On selfinjective artin algebras having nonperiodic generalized standard Auslander-Reiten components"Colloquium Mathematicum. in print.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] B. Kulshammer: "Some inequalities between invariants of blocks"Archiv der Mathematics (Basel). 79. 81-86 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yuji, Yoshino: "On degenenrations of Cohen-Macaulay modules"Journal of Algebra. 248. 272-290 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Andrzej Skowronski: "On invariability of selfinjective algebras of tilted type under stable equivalences"Proceeding of American Mathematical Society. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] Kunio Yamagata: "The rigid auto norphism group of a repetitive algebra"Proceedings of International Conference on Representation Theory of Algebras. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Otto Kerner: "Auslander-Reiten components containing cones"Algebras and Representation Theory. (印刷中).

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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