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. ]).

在2000年，获得以下结果：1。让m成为平滑的紧凑型歧管，而diff^k（m）是M上所有c^k diffemoritism的组。那么在diff^k（m）上存在一个概率度量μ，这是quasi-invariant在quasi-Invariant下，在平滑差异的左侧作用下是quasi-Invariant。在M上，平滑差异的平滑差异的每个连续统一表示在自然假设下都具有光滑载体的密集。3。上述结果自然扩展到非紧凑的M.SubSequient端序列，在2001年期间，我们考虑了上述结果和相关主题的直接问题，例如：1。从测量μ的自然表示不可约性。 2。μ本身的牙齿性。不幸的是，到目前为止，我们尚未为这些问题定义结论，也没有在量子力学中的第二个量化中出现在当前代数中的很好的应用。另一方面，我们考虑到拓扑组的电感限制，在此期间得到了公平的发展。我们现在正在争论拓扑空间的归纳限制，其直接产品和与代数相关的问题。到目前为止，获得了几个结果（请参阅[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 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。

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。

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: "Unitary representations and differential representations of the group of diffeomorphisms and its applications"Transactions of a Japanese-German Symposium. 319-333 (2000)

H.Shimomura：“微分同胚群的酉表示和微分表示及其应用”日德研讨会论文集。