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Analytic deformation of Poisson manifolds and noncominutative geometry

Research Project

Project/Area Number 13640208
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) OHYAMA Yoshiyuki  Tokyo Wemen's Christian University, Faculty of Science and Literature, Associate Professor of Mathematics, 文理学部, 助教授 (80223981)
NAKAMURA Yoshihiro  Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of Mathematics, 工学部, 助教授 (50155868)
ADACHI Toshiaki  Nagoya Institute of Technology, Faculty of Engineering, Associate Professor of Mathematics, 工学部, 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) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsPoisson manifold / symplectic manifold / analytic deformation / C^*-algebra / noncommutative geometry / 非可換幾何学 / 厳密量子化 / C^*-環 / 非可換多様体
Research Abstract

The aim of the project is to give a constructive proof of existence of analytic deformation of Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization(algebraic deformation) for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The relationship between algebraic deformation and analytic deformation is similar to the relationship between a formal power series and a smooth function that realizes the given formal power series.
Symplectic manifolds are special examples of Poisson manifolds, and its structures are well known. In a joint project with R. Nest of the University of Copenhagen and I.Peter of Munster University, we investigated symplectic manifolds and showed that any closed symplectic manifold has an analytic deformation provided that the second homotopy group is trivial. This result is published as "Strict quantization of symplectic manifolds (to appear in Letters hi Mathematical Physics)".
The second homotopy group of the 2-sphere is nontrivial. Thus the result above cannot be applied to the 2-sphere. In a joint project with C.L.Olsen of the State University of New York at Buffalo, we studied the 2-sphere. The 2-sphere possesses interesting Poisson structures besides the standard rotation invariant symplectic structure. We constructed an analytic deformation for a Poisson structure degenerate at the North and South poles. This result is published as "A new family of noncommutative 2-sphers".

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] T.Natsume, R.Nest: "Strict quantization of symplectic manifolds"Letters in Mathematical Physics. (掲載予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Natsume, C.L.Olsen: "A new family of noncommutative 2-spheres"Journal of Functional Analysis. (掲載予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda, K.Suzuki: "Characterization of totally geodesic Kahler immersions"Hokkaido Mathematical Journal. 31(3). 629-641 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda: "Characterization of space forms by circles in their geodesic spheres"Proceedings of Japan Academy, Ser. A Math. Sci.. 78(7). 143-147 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda, K.Suzuki: "A characterization of Veronese imbeddings into complex projective Spaces by circles"C. R. Math. Acad. Sci. Soc. R. Can. 24(2). 61-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda, M.Yamagishi: "Length spectrum of geodesic spheres in a non-flat complex space form"Journal of Mathematical Society of Japan. 54(2). 373-408 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 夏目利一, 森吉仁志: "作用素環と幾何学"日本数学会. 228 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Natsume and R.Nest: "Strict quantization of symplectic manifolds"Letters in Mathematical Pysics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Natsume and C.L.Olsen: "A new family of noncommutative 2-spheres"Journal of Functional Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda and K.Suzuki: "Characterization of totally geodesic Kahler immersions"Hokkaido Mathematical Journal. 31. 629-641 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi and S.Maeda: "Characterization of space forms by circles in their geodesic spheres"Proceedings og Japan Academy, Series A, Mathematical Sciences. 78. 143-147 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda and K.Suzuki: "Characterization of Veronese imbedding into complex projective spaces by circles"C.R.Mathematical Report, Academy of Science of Canada. 24. 61-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Adachi, S.Maeda and M.Yamagishi: "Length spectrum of geodesic spheres in a non-flat complex space form"Journal of Mathematical Society of Japan. 54. 373-408 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Moriyoshi and T.Natsume: "Operator Algebras and Geometry (in Japanese)"Mathematical Society of Japan. 228 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Natsume, R.Nest: "Strict quantization of symplectic manifolds"Letters in Mathematical Physics. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Natsume, C.L.Olsen: "A new family of noncommutative 2-spheres"Journal of Functional Analysis. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Adachi, S.Maeda, K.Suzuki: "Characterization of totally geodesic Kahler immersions"Hokkaido Mathematical Journal. 31(3). 629-641 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Adachi, S.Maeda: "Characterizations of space forms by circles in their geodesic spheres"proceedings of Japan Academy, Ser.A Math. Sci.. 78(7). 143-147 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Adachi, S.Maeda, K.Suzuki: "A characterization of Veronese imbeddings into complex projective Spaces by circles"C. R. Math. Acad. Sci. Soc. R. Can. 24(2). 61-66 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Adachi, S.Maeda, M.Yamagishi: "Length spectrum of geodesic spheres in a non-flat complex spaces form"Journal of Mathematical Society of Japan. 54(2). 373-408 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 夏目利一, 森吉仁志: "作用素環と幾何学"日本数学会. 228 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Natsume: "C^*-algebraic deformation and index theory"Mathematical Physics Studies. 23. 155-167 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Ohyama, K.Taniyama: "Vassiliev invariants of knots in a spatial graph"Pacific Journal of Mathematics. 200・1. 191-205 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Nakanishi, Y.Ohyama: "Knots with given finite type invariants and Ck-distance"Journal of Knot Theory and Its Ramifications. 10・7. 1041-1046 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 夏目利一, 森吉仁志: "作用素環と幾何学"日本数学会. 228 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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