2002 Fiscal Year Final Research Report Summary
Investigation of algebras of infinite representation type
| Project/Area Number |
12640014
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo University of Agriculture and Technology |
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Principal Investigator |
YAMAGATA Kunio Tokyo University of Agriculture and Technology, Department of Technology, Professor, 工学部, 教授 (60015849)
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| 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)
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| Project Period (FY) |
2000 – 2002
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| Keywords | finite dimensional algebra / representation / repetitive algebra / Frobenius algebra / module / socle deformation |
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| 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.
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