| Project/Area Number |
15540060
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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 |
Geometry
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| Research Institution | Saitama University |
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Principal Investigator |
MIZUTANI Tadayoshi Saitama University, Dept.of Math., Professor, 理学部, 教授 (20080492)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAMOTO Kunio Saitama University, Dept.of Math., Professor, 理学部, 教授 (70089829)
NAGASE Masayoshi Saitama University, Dept.of Math., Professor, 理学部, 教授 (30175509)
FUKUI Toshizumi Saitama University, Dept.of Math., Professor, 理学部, 教授 (90218892)
SAKAI Fumio Saitama University, Dept.of Math., Professor, 理学部, 教授 (40036596)
SHIMOKAWA Koya Saitama University, Dept.of Math., Associate Professor, 理学部, 助教授 (60312633)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Poisson manifold / Lie algebroid / Schouten bracket / multi-vector field / singular foliation / almost Dirac structure / Dirac structure / ディラック構造 |
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| Research Abstract |
One of the themes of our research throughout the project was the Lie algebroid which is naturally associated with a 2-vector field. More precisely, let π denote a 2-vector field on M. π is regarded as a bundle map of T*M to TM. The image D of π is a distribution (plane field). Complete integrability of D is investigated by the formula
π({α,β})=[π(α),π(β)]・1/2 [π,π] (α,β).
In fact, D is integrable if and only if Ker π⊂Ker[π,π] holds. Moreover, Ker[π,π] forms a Lie algebroid if it is a subbundle of T*M of constant rank. We generalized this argument on the cotangent bundles to the cases of general Lie algebroids. Further we treated the cases of deformed brackets by closed 1-form.
Recently, we showed that the problems are well-treated and proofs are simplified greatly by working in the framework of Dirac structures which was introduced'by Courant and Weinstein before. These results can be found in the following joint papers with K.Mikami.
1)K.Mikami and T.Mizutani ;
Lie algebroid associated with an almost Dirac structure., Travaux mathematiques Vol 16,255-264(2005)
2)K.Mikami and T.Mizutani ;
A Lie algebroid and a Dirac structure associated to an almost Dirac structure (To appear)
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