1998 Fiscal Year Final Research Report Summary
Infinite Dimensional Representation, Measure Theory and Related Topics
| Project/Area Number |
09640171
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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 |
解析学
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| Research Institution | Fukui University |
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Principal Investigator |
SHIMOMURA Hiroaki Fukui University, Faculty of Education, Professor, 教育学部, 教授 (20092827)
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| Co-Investigator(Kenkyū-buntansha) |
MIKAMI Shunsuke Fukui Medical University, Faculty of Medicine, Professor, 医学部, 教授 (00126640)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Manifold / Group of Diffeomorphism / Unitary Representation / Differential Representation / Infinite Dimensional Lie group / Infinite Dimensior / Linearity / 1-Cocycle |
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| Research Abstract |
Between these two years I contained to study on unitary representations of the group of. diffeomorphisms with compact support Diff_0(M) or of its subgroups on smooth manifolds M.It is known that these groups are infinite dimensional Lie groups, whenever M is compact. Hence there is possibility to analyze these representations with the Lie algebraic method. Under these considerations I have obtained the following results for reducibility our unitary representations to the linear one.
1 The linearlity is assured by a formula which corresponds to the Campbell-Hausdorff formula on the usual Lie group. (In our case, the formula comes from an evaluation for the behavior of solutions of some autonomus differential equations)
2. A chracterization of the subgroup generated by the image of Lie algebra by the exponential mapping.
For the above problem I have seen that it is no problem to proceed our theories, for example in the case of Diff_0(M), the subgroup is dense in the connected component of the neutral element.
3. Lastly, for the problem of rich existence of C^*-vectors I am continuing to discuss it now, of course on infinite dimensional representations.
Moreover I applied the above results to 1-cocycles in terms of Diff_0(M) and obtained some fundamental results. In particular the cocycle form has a close connection with the geometrical structure on M.
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Research Products
(12 results)
