2022 Fiscal Year Final Research Report
Infinite dimensional geometry of Kac-Moody groups and integrable systems
| Project/Area Number |
20K22309
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Osaka Metropolitan University (2022)
Osaka City University (2020-2021) |
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Principal Investigator |
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| Project Period (FY) |
2020-09-11 – 2023-03-31
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| Keywords | カッツ・ムーディ幾何学 / 無限次元部分多様体 |
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| Outline of Final Research Achievements |
We obtained the following results which deepen the relation between infinite dimensional submanifold geometry and Kac-Moody geometry: We gave an explicit formula for orbits of path group actions given as isotropy representations of affine Kac-Moody symmetric spaces. We newly defined a certain isomorphism of path spaces and unified all known computational results of principal curvatures of proper Fredholm submanifolds. We showed that the isomorphism corresponds to an isomorphism of affine Kac-Moody symmetric spaces of group type. As an application of these we constructed many examples of infinite dimensional austere submanifolds.
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| Free Research Field |
微分幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
カッツ・ムーディ代数は，無限の対称性を記述する言語として，数学や物理の様々な分野で現れる．しかし，無限である故にその取り扱いが難しく，特にカッツ・ムーディ群は，これまで避けられることが多かった．本研究では，無限次元部分多様体幾何学とカッツ・ムーディ幾何学の観点から，カッツ・ムーディ代数・群や関連分野に対する理解を深めることができた．これは数学や物理の基礎研究において，1つの基盤となる意義のある成果であると考えている．
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