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2000 Fiscal Year Final Research Report Summary

Quantization of Poisson manifolds and noncommutative geometry

Research Project

Project/Area Number 11640198
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionNagoya Institute of Technology

Principal Investigator

NATSUME Toshikazu  Nagoya Institute of Technology, Faculty of Engineering, Professor of Mathematics, 工学部, 教授 (00125890)

Co-Investigator(Kenkyū-buntansha) NAKAMURA Yoshihiro  Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of Mathematics, 工学部, 助教授 (50155868)
OHYAMA Yoshiyuki  Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of Mathematics, 工学部, 助教授 (80223981)
ADACHI Toshiaki  Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of Mathematics, 工学部, 助教授 (60191855)
MORIYOSHI Hitoshi  Keio University, Faculty of Science and Engineering, Associate Professor of Mathematics, 理工学部, 助教授 (00239708)
Project Period (FY) 1999 – 2000
KeywordsPoisson manifold / deformation quantization / C^*-algebra / strict deformation quantization / Strict quantization
Research Abstract

In a joint project with R.Nest of the University of Copenhagen and I.Peter of the University of Munster the pricipal investigator showed that under a topological condition every closed symplectic manifold has a strict quantization. Strict quantization is an analytic deformation theory. An algebraic deformation theory (existence of deformation quantization) has been known since 80's.
The aim of the project is to show existence of strict quantizations for Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The project is divided into three steps. The first step is to re-examine the existence proof of strict quantization for symplectic manifolds, in order to have a deep understanding of mechanism of existence. In particular, re-examination of the proof by B.Fedosov, which played a crucial role in our proof, of existence of deformation qu … More antization is an important step. The second step is to understand the existence proof of deformation quantization for Poisson manifolds and to rewrite it from the viewpoint of Fedosov. The last step involves actual construction of strict quantization.
Through quite a few discussions with Nest, the mechanism of existence became fairly clear, and we obtained a refined version of our result. Thanks to a recent appearance of a simpler proof of existence of deformation quantization for Poisson manifolds than Kontsevich's, we have a prospect to achieve the second step.
While working on the project discussed above, in a joint project with C.L.Olsen of the State University of New York at Buffalo, the principal investigator worked on the cases that are not covered by the results with Nest and Peter. In particular, we showed that the 2-sphere with a specific Poisson structure has a strict quantization. In the process to construct strict quantization we obtained new "noncommutative 2-spheres". These C^*-algebras are new examples of noncommutative Poisson manifolds.
As explained above, unfortunately we could not achieve the goal of the project, i.e. the existence of strict quantizations for poisson manifolds. We certainly intend to continue working on the project. We will hopefully complete the project within a year or so. Less

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] T.Natsume,R.Nest: "Topological approach to surfaces"Communications in Mathematical Physics. 202. 65-87 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Natsume: "C^*-algebraic deformation guontization of closed Riemann surfaces"Proceedings of the SNB-Workshop on C^*-algebras. 142-150 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Natsume: "C^*-algebraic deformation and index theory"Proceedings of Workshop "Quantization", Shona Kokusainam. (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 夏目利一: "トポロジストの為の作用素環論入門"日本数学会 日本語メモワール:作用素環と幾何学. 2. (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Adachi: "Spaceforms from the viewpoint of their geodesic spheres"Bulletin of the Australian Mathematical Society. 62. 205-210 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohyama: "Web diagrams and realization of Vassilier invariants by Knots"Journal of Knot Theory and its Ramifications. 9. 693-701 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Natsume and R.Nest: "Topological approach to quantum surfaces"Communications in Mathematical Physics. 202. 65-87 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Natsume: "C^*-algebraic deformation quantization of closed Riemann surfaces"Proceeding of the SNB-Workshop on C^*-algebras, Muenster, Germany. 142-150 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Natsume: "C^*-algebraic deformation and index theory"Proceeding of Workshop on Quantizations, Shonan Kokusaimura. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Natsume: "Operator Algebras for Topologists (in Japanese)"Japanese Mathematical Society Memoir in Japanese vol. 2 "Operator Algebras and Geometry". (in print.).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Adachi: "Space forms from the viewpoint of their geodesic sphere"Bulletin of the Australian Mathematical Society. 62. 205-210 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Ohyama: "Web diagrams and realization of Vassiliev invariants by knots"Journal of Knot Theory and its Ramifications. 9. 693-701 (2000)

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

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Published: 2002-03-26  

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