2018 Fiscal Year Final Research Report
Studies of cohomology groups associated with Poisson structures
| Project/Area Number |
26400063
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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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| Research Field |
Geometry
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| Research Institution | Akita University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Keywords | ポアソン構造 / リー代数・リー超代数 / ヤング図形 / (コ)ホモロジー群 / ゲルファント・フックス理論 / 荷重 / オイラー数 / ベッチ数 |
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| Outline of Final Research Achievements |
The set of Hamiltonian vector fields from a Poisson structure consists a Lie algebra. Gel'fand-Fukｓ theory dealt with (co)chain complex and (co)homology groups of those Lie algebras and reduced general discussions to finite dimensional ones by using "weight". In our research project, we got results in 3 branches.
(1) In the relative GF-cohomology group of formal Hamiltonian vector fields on the 2-plane of weight 24, we identified all the Betti numbers after a long calculation of more than five years. There are 3 independent cycles in degree 7. (2) We modified the weight and developed Gel'fand-Fuks type theory for not only symplectic but also Poisson structures of homogeneous polynomial coefficients on vector space. (3) By Schouten bracket, the set of multivector fields of various degrees forms a Z-graded Lie superalgebra. We introduced a notion of "double weight" and obtained examples of polynomial coefficient multivector fields on n-dimensional vector space.
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| Free Research Field |
ポアソン幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
斜交平面の形式的ハミルトン場の荷重 24 ゲルファント・フックスコホモロジー群について、非自明なベッチ数の在り場所を特定出来、非自明な物の幾何学的性質の研究という新たな課題を見いだした意義は大きい。斜交空間でのゲルファント・フックス理論をポアソン空間のハミルトンベクトル場に対しても一般化した事で新たな研究の展開が期待される。
リー代数(環)の(コ)ホモロジー理論の類推としてのリー超代数の(コ)ホモロジー理論が成立し二重荷重(double weight)で有限化出来る例の存在も示し、ポアソン構造そのものの研究に寄与するとの見方を得た事は今後のポアソン幾何研究に貢献すると考える。
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