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Research of non-commutative differential geometry

Research Project

Project/Area Number 09440041
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionScience University of Tokyo

Principal Investigator

OMORI Hideki  Science University of Tokyo, Dept.Math. Science and Technology, Prof., 理工学部, 教授 (20087018)

Co-Investigator(Kenkyū-buntansha) OKA Hasatoshi  Science University of Tokyo, Science and Technology, Dept.Math.Prof., 理工学部, 教授 (70120178)
SHOJI Toshiaki  Science University of Tokyo, Science and Technology, Dept.Math.Prof., 理工学部, 教授 (40120191)
ARAKI Fujihiro  Science University of Tokyo, Science and Technology, Dept.Math.Prof., 理工学部, 教授 (20027361)
YOOHIOKA Akira  Science University of Tokyo, Science and Technology, Dept.Math.Asso.Prof., 理工学部, 助教授 (40200935)
FURUTANI Kense  Science University of Tokyo, Science and Technology, Dept.Math.Prof., 理工学部, 教授 (70112901)
原 民夫  東京理科大学, 理工学部, 講師 (10120205)
田宮 高紀  東京理科大学, 理工学部, 講師 (60183472)
大槻 舒一  東京理科大学, 理工学部, 教授 (80112895)
小林 嶺道  東京理科大学, 理工学部, 教授 (70120186)
Project Period (FY) 1997 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥7,200,000 (Direct Cost: ¥7,200,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1998: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsnon commutative geometry / star-product (*-product) / non commutative differential geometry / *-積 / 水一積 / 幾何学的量子化
Research Abstract

The field of deformation quatization is one of the most active, intersecting area of mathematics and physics.
There are several kernel places in the world, working to making quantum calculus. It was very important and fruitful to discuss with many people who are interested in this field.
In this period of research, the following are discovered :
The notion of μ regulated algebras can be the most fundamental notion for the quantum calculus, and for the non-commutative differential geometry, even for the case the Plank constant h is viewed as a genuine parameter moving in positive reals.
In such system, we found special elements, that play the same role as vacuum in the classical quantum theory.
In the case that h is viewed as a positive real parameter, it is crucial to fix the product formula, for the star-exponential functions of quadratic forms.
Here we found strange phenomena that quadratic forms with non-vanishing discriminant have two different inverses and therefore the associativity breaks down.
By this phenomenon, we are forced to break some symmetry in order to keep associativity. Infact, we have to use SL (2 ; R)-symmetry instead of SL(2 ; C)-symmetry.

Report

(5 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (28 results)

All Other

All Publications (28 results)

  • [Publications] 大森英樹: "Groups of quantum volume preserving diffeomorphisms and their Beregin representation"Analysis on infinite dimensional Lie groups (論文集) World Scientific. 337-354 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "On global hypoelliptiaty on compaet manifolds"Hokkaido Mathematical Journal. 28. 613-633 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "Noncommutative world, and its geometric picture"A.M.S.translation of Sugaku expositions. 13・2. 143-171 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "Deformation quantization of Frechet-Poirson algebras"Mathematical Physics Studies (論文集). 22. 233-246 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "Singular system of exponential functions"Noncommutative Differential Geometry (論文集). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "One must break symmetry in order to keep associativity"Banach center publications. 44・. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "無限次元Lie群論"アメリカ数学会. 415 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Noncommutative 3-sphere"J.Math.Soc.Japan. vol 50 no 4. 945-943 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Poincae-Cartan class and deformation quantization."Commun.Math.Phys.. 194. 207-230 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Groups of quantum volume pre-serving diffeomorphisms and their Berezin representation."in the book Analysis on infinite-dimensional Lie groups and algebras, ed. H.Heyer, J.Marion, World Scientific. 337-354 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Deformation quantization of the Poisson algebra of Laurent polynomials"Lett.Math.Pysics.. 46. 171-180 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, T.Kobayashi: "On global hypoellipticity on compact manifolds"Hokkaido Math.J. 28. 613-633 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Deformation quantization of Frechel-Poisson algebras, Convergence of the Moyal product"in the book Mathematical Physics Studies, 22 Kluwer Academic Publishers. 233-246 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Singular system of exponential functions"in Noncommutative Differential Geometry and its Applications to Physics, Kluwer Academic Publishers.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 大森英樹: "Noncommutative world, and its geometric picture"A.M.S.translation of Sugaku expositions. 13・2. 143-171 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 大森英樹: "Deformation quantization of Fiechet-Poisson algebras,"Mathematical Physics Studies(論文集). 22. 233-246 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 大森英樹: "Singular system of exponential functions"Noncommutative Differential Geometry(論文集). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 大森英樹: "Deformation quantization of Fiechet-Poisson algebras of Heisenberg type"Differential geometry(論文集).

    • Related Report
      2000 Annual Research Report
  • [Publications] 大森英樹: "One must break symmetry in order to keep associativity"Banach center publications. 44. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 大森英樹: "Global Hypoellipticity of subelliptic operators on closed manifolds"Hokkaido Mathematical Journal. 28・3. 613-633 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大森英樹: "Noncommutative woeld,and its geometrical picture"A.M.S. translation of Sugaku expositions. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大森英樹: "Deformation quantization of Frechet-Poisson algebras"Lett.Math.Physics. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大森,前田,宮崎,吉岡: "Poincar'e-Cartan class and deformation quautization of Kahler wamifods" Communications in Mathematical Physics. 194. 207-230 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大森,前田,宮崎,吉岡: "Noncommutative 3-sphere:A model of coucommutative coutact algebras" Journal of Mathematical Society of Japan. 50-4. 915-943 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 荒木.Flato,Sternheimer: "Some infinite dimewional algebra arizing in spin systems and in particle physics" Letters in Mathematical physics. 43. 155-171 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大森英樹: "非可換の世界と,幾何学的描像" 数学. 50・1. 12-28 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.OMORI: "NONCOMMUTATIVE 3-SPHERE" J.Math.Soc.Japan. 50・4(発表予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.OMORI: "Poincare-Cartan Class and Deformation Quantization of Kahler Mauifolds" Commun.Math.Phys.155(発表予定). (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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