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2002 Fiscal Year Final Research Report Summary

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)
Project Period (FY) 2000 – 2002
Keywordsfinite dimensional algebra / representation / repetitive algebra / Frobenius algebra / module / socle deformation
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.

  • Research Products

    (11 results)

All Other

All Publications (11 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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 2004-04-14  

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