2007 Fiscal Year Final Research Report Summary
STUDY ON FULL MATRIX ALGEBRAS WITH STRUCTURE SYSTEMS
| Project/Area Number |
18540011
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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 | University of Tsukuba |
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Principal Investigator |
FUJITA Hisaaki University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor (60143161)
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| Co-Investigator(Kenkyū-buntansha) |
増岡 彰 筑波大学, 大学院・数理物質科学研究科, 准教授 (50229366)
星野 光男 筑波大学, 大学院・数理物質科学研究科, 講師 (90181495)
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| Project Period (FY) |
2006 – 2007
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| Keywords | matrix ring / Frobenius algebra / tiled order / global dimension / characteristic of field |
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| Research Abstract |
Concerning full matrix algebras with structure systems, we obtained the following research results.
1. For an arbitrarily given structure system, we defined its (0, 1)-limit full matrix algebra. Let A be an n × n full matrix algebra with a structure system. In the case of n=3, we proved that A is isomorphic to its (0, 1)-limit and we obtained the list of five isomorphism classes. For each n≧4, we constructed a family of mutually non-isomorphic Frobenius full matrix algebras having the same (0, 1)-limit, which are parameterized by elements of a base field. Considering the set of structure systems as an algebraic variety, we showed that each isomorphism class of full matrix algebras corresponds to a G-orbit of the variety of structure systems, where G is a certain group acting on the variety.
2. Concerning tiled orders over a discrete valuation ring, a question was posed in a paper of Fujita (J. Algebra, 2002), that is, research, we found a counterexample to this question. We note that the question was one of the motivations to introduce full matrix algebras with structure systems, and that the counterexample had not been expected before. Our example also shows that the characteristic of a base filed plays an important role in the study of full matrix algebras with structure systems.
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