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Infinite Dimensional Representation, Measure Theory and Related Topics

无限维表示、测度论及相关主题

基本信息

批准号：
09640171
负责人：
SHIMOMURA Hiroaki
金额：
$ 0.51万
依托单位：
Fukui University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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.

在这两年期间，我致力于研究群的统一表示。具有紧支持 Diff_0(M) 或其子群在光滑流形 M 上的微分同胚。众所周知，只要 M 是紧的，这些群就是无限维李群。因此可以用李代数方法分析这些表示。在这些考虑下，我获得了以下关于我们的酉表示可还原为线性表示的结果。1线性度由对应于通常李群上的 Campbell-Hausdorff 公式的公式保证。（在我们的例子中，该公式来自对一些自治微分方程的解的行为的评估）2。通过指数映射对李代数图像生成的子群进行表征。对于上述问题，我发现继续我们的理论是没有问题的，例如在 Diff_0(M) 的情况下，子群在中性元素的连通分量中是稠密的。 3．最后，对于C^*向量的丰富存在性问题我现在继续讨论，当然是在无限维表示上。此外，我将上述结果应用到Diff_0(M)的1-cocycles上，并得到了一些基本结果。特别是余循环形式与M上的几何结构有着密切的联系。