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2001 Fiscal Year Final Research Report Summary

Infinite Dimensional Representations and Related Topics

Research Project

Project/Area Number 12640164
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionFukui University

Principal Investigator

SHIMOMURA Hiroaki  Fukui University Faculty of Enginnering, Professor, 工学部, 教授 (20092827)

Co-Investigator(Kenkyū-buntansha) YASUKURA Osami  Fukui University Faculty of Enginnering, Associate Professor, 工学部, 助教授 (00191122)
ONODA Nobuharu  Fukui University Faculty of Enginnering, Professor, 工学部, 教授 (40169347)
MIKAMI Shunsuke  Fukui Medical University, Factory of Medicine, Professor, 医学部, 教授 (00126640)
Project Period (FY) 2000 – 2001
KeywordsManifold / Difisomorphism / Quasi-Invariant Measure / Unitary Representation / Smooth Vector / Inductive Limit / 直積構造
Research Abstract

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. ]).

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] H.Shimomura: "Unitary representations and differential representations of the group of diffeomorphisms and its applications"Proceedings of JSPPS-DFG Japan-Germany Joint Seminar, IDHA. 319-333 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Hirai et al.: "On inductive limit topological algebraic structures in relation to product topologies"Proceedings of JSPPS-DFG Japan-Germany Joint Seminar, IDHA. 177-191 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Shimomura: "Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations"Journal of Functional Analysis. 187. 406-441 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Hirai et al.: "Inductive limit topologies, their direct products and problems related algebraic structures"J. Math. Kyoto Univ.. 41. 475-505 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H. Shimoinura: "Unitary representations representations of the group of diffeomorphisms and its applications"Proceedings of JPSP-DFG Japan-Germany Joint Seminar. IDHA. 319-333 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Hirai et al.: "On inductive limits of topological algebraic structures in relation to product topologies"Proceedings of JPSP-DFG Japan-Germany Joint Seminar. IDHA. 177-191 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. Shimomura: "Quasi-invariant measures on the group ofi diffeomorphisms and smooth vectors of unitar representations"Journal of Functional Analysis. 187. 406-441 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Hirai et al.: "Inductive limit topologies, their direct products ana prooiems related algebraic structures"Journal of Mathematics Kyoto University. 41. 475-505 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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