Infinite Dimensional Representation, Measure Theory and Related Topics

无限维表示、测度论及相关主题

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640171/
• 关键词: Manifold; Group of Diffeomorphism; Unitary Representation; Differential Representation; Infinite Dimensional Lie group; Infinite Dimensior; Linearity; 1-Cocycle; コニタリ表現; 弥形性; 帰納極限; 位相群

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.

在这两年中，我开始研究群的酉表示。光滑流形M上紧支持Diff_0(M)或其子群的微分同态，已知当M紧时，这些群是无穷维李群。因此，有可能用李代数方法来分析这些表示。在这些考虑下，我得到了以下结果，可以将我们的幺正表示约化为线性表示线性性由一个公式来保证，该公式与一般李群上的Campbell-Hausdorff公式相对应。（在我们的例子中，这个公式来自于对一些自主微分方程解的行为的评估）用指数映射刻画李代数象所产生的子群。对于上述问题，我已经看到，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密集的。最后，对于C^*向量的丰富存在性问题，我现在继续讨论它，当然是在无限维表示上。并将上述结果应用到关于Diff_0(M)的1环上，得到了一些基本结果。特别是循环形式与M上的几何结构有密切的联系。

项目成果

H.Shimomura: "Unitary representations and 1-cocycles on the group of diffeomorphisms" RIMS.kokyuroku. (近刊).

H.Shimomura：“微分同胚群上的幺正表示和 1-余循环”RIMS.kokyuroku（即将出版）。

H.Shimomura: "Relations between unitary representations of diffeomorphism groups and those of the infinite symmetric group or related permutation groups (with T.Hirai)" J.Math.Kyoto Uinv.37. 261-316 (1997)

H.Shimomura：“微分同胚群的酉表示与无限对称群或相关排列群（与 T.Hirai）的酉表示之间的关系”J.Math.Kyoto Uinv.37。

H.SHIMOMURA: "CANONICAL Representations Generated by translctconally quesi-inva" riant measnres,PUBL.RIMS.Kyoto Univ.32.No4. 633-669 (1996)

H.SHIMOMURA：“由 translctconally quesi-inva 生成的规范表示”riant measnres，PUBL.RIMS.Kyoto Univ.32.No4。

下村 宏彰: "1-cocycles on infinite dimensional spaces" 数理解析研究所講究録. 1017. 116-123 (1997)

Hiroaki Shimomura：“无限维空间上的 1-cocycles”数学分析研究所的 Kokyuroku。1017. 116-123 (1997)

H.Shimomura: "1-cocycles on the group of diffeomorphisms." J.Math.Kyoto Univ.38(in press).

H.Shimomura：“微分同胚群上的 1-余循环。”