Infinite Dimensional Representation, Measure Theory and Related Topics
Project/Area Number  09640171 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Fukui University 
Principal Investigator 
SHIMOMURA Hiroaki Fukui University, Faculty of Education, Professor, 教育学部, 教授 (20092827)

CoInvestigator(Kenkyūbuntansha) 
MIKAMI Shunsuke Fukui Medical University, Faculty of Medicine, Professor, 医学部, 教授 (00126640)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥400,000 (Direct Cost : ¥400,000)
Fiscal Year 1997 : ¥400,000 (Direct Cost : ¥400,000)

Keywords  Manifold / Group of Diffeomorphism / Unitary Representation / Differential Representation / Infinite Dimensional Lie group / Infinite Dimensior / Linearity / 1Cocycle / 多様体 / 微分同相群 / ユニタリ表現 / 微分表現 / 無限次元Lie群 / 無限次元Lie環 / 線形性 / 1cocycle / コニタリ表現 / 弥形性 / 帰納極限 / 位相群 
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 CampbellHausdorff 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 1cocycles 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. 更に以上の成果を群Diff_0(M)に関する1cocycleに応用し、幾つかの基本的結果を得た。特に、lcocycleの形はMの幾何学的形状と密接な関係があることがわかった。

Report
(4results)
Research Output
(18results)