| Project/Area Number |
18K03203
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Ibaraki University |
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Principal Investigator |
Yutaka Yoshii 茨城大学, 教育学部, 准教授 (90613141)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2019: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2018: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
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| Keywords | 代数群の表現論 / モジュラー表現論 / 原始冪等元 / 射影直既約加群 / 有限次元多元環 / 有限次元多元環の表現論 |
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| Outline of Final Research Achievements |
Through my previous research, certain elements (denoted as B(ε)(a,j) hereafter) including primitive idempotents in the hyperalgebra Dist(G_r) of the r-th Frobenius kernel G_r of the algebraic group G=SL(2,k) over a field k of positive characteristic have been obtained. Using these elements, I succeeded in constructing certain bases and generating sets of the Jacobson radical for Dist(G_r). Moreover, I succeeded in constructing generators for the hyperalgebra Dist(G_r) of the Frobenius kernel G_r for a general simply connected simple algebraic group G, and further, using a linear map associated with the Frobenius map, it was found that several linear isomorphisms are determined by the product of rings for Dist(G_r) and its major subalgebras.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究において、申請者の先行研究によって得られた、G=SL(2,k)の第r Frobenius核G_rの超代数Dist(G_r)の原始冪等元を含む元B(ε)(a,j)たちを用いて、Dist(G_r)における環論的な性質をいくらか記述することに成功した。学術的意義としては、今回の研究成果が一般の半単純代数群G（およびその第r Frobenius核G_r）の表現論における新たな研究手法を確立するための第一歩となり得ること、さらにその結果や考え方が関連する代数に応用できる可能性があることである。
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