Infinite Dimensional Representations and Related Topics

无限维表示及相关主题

• 批准号: 12640164
• 负责人: SHIMOMURA Hiroaki
• 金额: $ 0.26万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640164/
• 关键词: Manifold; Difisomorphism; Quasi-Invariant Measure; Unitary Representation; Smooth Vector; Inductive Limit; 直積構造; 準不変測度; 滑らかなベクトル; Diffeomorphism Group; Unitary Representation; Quasi-Invariant Measure; Smooth Vector; Regular Representation

In 2000 the following results are obtained : 1. Let M be a smooth compact manifold and Diff^k (M) be the group of all C^k diffeomorphisins on M. Then there exists a probability measure μ on Diff^k (M) that is quasi-invariant under the left action of the smooth diffeomorphisms.2. Every continuous unitary representation of the group of smooth diffeoinorphisros on M has a dense setof the smooth vectors under a natural hypothesis.3. The above result is naturally extended to non compact M.Subsequently during the period of 2001 we considered direct questions of the above results and related topics, for example : 1. Irreducibility of the natural representation derived from the measure μ. 2. Ergodicity of μ itself.Unfortunately we have neither definite conclusions for these questions up to now nor nice applications to current algebra appeared in the second quantization in quantum mechanics.On the other hand our considerations to inductive limit of topological groups are fairly developed in this period. We are now arguing inductive limit of topological spaces, their direct products and problems related algebraic structures. Several results are obtained till now (See [T. Hirai et al. ]).

2000年取得了如下成果：1.设M是光滑紧致流形，Diff^k（M）是M上所有C^k同态的群.则在Diff^k（M）上存在一个概率测度μ，它在光滑同态的左作用下是拟不变的.在自然假设下，M上光滑非同构群的每个连续酉表示都有光滑向量的稠密集.上述结果自然地推广到非紧M。随后在2001年期间，我们考虑了上述结果的直接问题和相关的主题，例如：1。从测度μ导出的自然表示的不可约性。2.μ自身的遍历性.遗憾的是，这些问题至今还没有明确的结论，也没有很好地应用于量子力学二次量子化中出现的当前代数.另一方面，我们对拓扑群归纳极限的考虑在这一时期得到了相当的发展.本文主要讨论拓扑空间的归纳极限、直积以及与代数结构有关的问题。到目前为止已得到了一些结果（见[T. Hirai等人]）。

项目成果

T.Hirai,H.Shimomura, et al.: "On inductive limits of topological algebraic structures in relation to the product topologies"Transactions of a Japanese-German Symposium. 177-191 (2000)

T.Hirai、H.Shimomura 等人：“论与乘积拓扑相关的拓扑代数结构的归纳极限”日德研讨会汇刊。

H.Shimomura: "Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations"Journal of Functional Analysis. 187. 406-441 (2001)

H.Shimomura：“微分同胚群和酉表示的平滑向量的准不变测度”泛函分析杂志。

H. Shimomura: "Quasi-invariant measures on the group ofi diffeomorphisms and smooth vectors of unitar representations"Journal of Functional Analysis. 187. 406-441 (2001)

H. Shimomura：“i 微分同胚组的准不变测度和幺正表示的平滑向量”函数分析杂志。

T.Hirai et al.: "Inductive limit topologies, their direct products and problems related algebraic structures"J. Math. Kyoto Univ.. 41. 475-505 (2002)

T.Hirai 等人：“归纳极限拓扑、它们的直积以及与代数结构相关的问题”J.

H.Shimomura: "Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations"Journal of Functional Analysis. vol.187 No.2. 406-441 (2001)

H.Shimomura：“微分同胚群和酉表示的平滑向量的准不变测度”泛函分析杂志。