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A study of quantum groups by using the theory of unbounded operator algebras

Research Project

Project/Area Number 10640222
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionFukuoka Universityq

Principal Investigator

KUROSE Hideki  Fukuoka Univ., Fac. Sci., Professor, 理学部, 教授 (00161795)

Co-Investigator(Kenkyū-buntansha) NAKAGAMI Yoshiomi  Yokohama City Univ., Dept. Sci., Prof., 理学部, 教授 (70091246)
ARAI Asao  Hokkaido Univ., Fac. Sci., Professor, 理学部, 教授 (80134807)
INOUE Atsushi  Fukuoka Univ., Fac. Sci., Professor, 理学部, 教授 (50078557)
OGI Hidekazu  Fukuoka Univ., Fac. Sci., Assistant, 理学部, 助手 (30248471)
NAKAZATO Hiroshi  Hirosaki Univ., Fac. Sci. & Tech., Prof., 理工学部, 教授 (10188922)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1998: ¥1,600,000 (Direct Cost: ¥1,600,000)
Keywordsquantum group / Hopf algebra / Hopf *-algebra / locally compact quantum group / quantum double / double group construction / Woronowicz unbound operator algebra / unbound operator algebra / ホップ*-代数 / Doubule Group Construction / 局所インパクト量子群
Research Abstract

For Hopf (*-) algebras A and B, using an exchanging map S : B 【cross product】 A → A 【cross product】 B we defined a twisted tensor product of those. We showed that twisted tensor product is not but the bicrossed product of a matched pair of the Hopf (*-) algebras A and B and that the quantum double is a particular example of twisted tensor products. For any coquasitriangular Hopf (*-) algebra A, which is not necessarily finite dimensional, we defined a (*-) algebra homomorphism from quantum double of A and itself into the tensor product of A and its quantum enveloping algebra and showed that it is an isomorphism if A is factorizable. This is a generalization of a result by S. Majid for finite dimensional Hopf algebras. Passing through the above homomorphism, for any coquasitriangular Hopf (*-) algebra A, we prove that the quantum double of A and itself is paired with the quantum enveloping algebra of A and the opposite Hopf (*-) algebra of A and that the pairing nondegenerate iff A is factorizable. The result implies that the quantum enveloping algebra of the complex quantum group of a compact quantum Lie group A corresponds to the quantum double of the quantum enveloping algebra of A and the opposite Hopf algebra of A. The quantum Lorentz algebra is an example. We discussed about the modular theory of compact quantum groups at the level of Hopf *- algebra and defined the notion of algebraic Woronowicz algebras. Taking the dual we could think of the modular theory for discrete quantum groups as multiplier Hopf algebras. The modular structure is further defined for the quantum groups which are described as the quantum double or the double group of compact or discrete quantum groups. In the regular bounded representations of algebraic Woronowicz algebras, we got the corresponding Woronowicz algebras as those weak closures.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (30 results)

All Other

All Publications (30 results)

  • [Publications] H. Kurose, Alfons Van Daele and Yinhuo Zhang: "Corepresentation theory of multiplier Hopf algebras II"International Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideki Kurose: "Quantum Doubles of Hopf *-algebras"RIMS Kokyuroku, Kyoto Univ.. 1110. 68-76 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Jean-Pierre Antoine, Atsushi Inoue and Hidekazu Ogi: "Standard generalized vectors and *-automorphism groups of partial O^*-algebras"Reports on Mathematical Physics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi Inoue, Witold Karwowski and Hidekazu Ogi: "Standard weights on algebras of unbounded operators"Journal of Math. Soc. Japan. 51. 911-936 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Subhash J. Bhatt, Atsushi Inoue and Hidekazu Ogi: "Admissibility of Weights on Non-normed *-Algebras"Transactions of the American Mathematical Society. 351(11). 4629-4656 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Subhash J. Bhatt, Atsushi Inoue and Hidekazu Ogi: "On C^*-spectral algebras"Rendiconti of the Circolo Matematico di Palermo, Serie II. 56. 207-213 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Asao Arai: "Strong anticommutativity of Dirac operators on boson-fermion Fock spaces and representations of a supersymmetry algebra"Mathematische Nachrichten. 207. 61-77 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Asao Arai: "Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials : Canonical commutation relations, quantum algebras"Journal of Mathematical Physics. 39. 2476-2498 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Nakazato: "Quintic elliptic curves arising in numerical range"Bulletin Sci. Tech. Hirosaki Univ.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hidekazu Ogi: "Unbounded C^*-seminorms and *-representations of *-algebras"RIMS Kokyuroku, Kyoto Univ.. 1077. 52-61 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi Inoue: "Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture notes in Mathematics 1699"Springer. 241 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Kurose, Alfons Van Daele and Yinho Zhang: "Corepresentation theory of multiplier Hopf algebras II,"International Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideki Kurose: "Quantum Doubles of Hopf *-algebras"RIMS Kokyuroku, Kyoto Univ.. 1110. 68-76 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Jean-Pierre Antoine, Atsushi Inoue and Hidekazu Ogi: "Standard generalized vectors and *-automorphism groups of patient OィイD1*ィエD1-algebras"Reports on Mathematical Physics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi Inoue, Witold Karwowski and Hidekazu Ogi: "Standard weight on algebras of unbounded operators"Journal of Math. Soc. Japan. 51. 911-936 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Subhash J. Bhatt, Atushi Inoue and Hidekazu Ogi: "Admissibility of Weight on Non-normed *-Algebras"Transactions of the American mathematical Society. 351(11). 4329-4356 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Subhash J. Bhatt, Atushi Inoue and Hidekazu Ogi: "On C*-spectral algebra"Rendiconti of the Circolo Mathematico di Palermo, Serie II. 56. 207-213 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Asao Arai: "Strong anticommutativity of Dirac operators boson-fermion Fock spaces and representation of a supersymmetry algebra"Mathematische Nachrichten. 207. 61-77 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Nakazato: "Quintic elliptic curves arising in numerical range"Bulletin Sci. Tech. Hirosaki Univ.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hidekazu Ogi: "Unbounded CィイD1*ィエD1-seminorms and *-representations of *-algebras"Rmk Kokyuryoku, Kyoto Univ.. 1077. 52-61 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi Inoue, Tomita-Takasaki: "Thoory in Algebras of Unbounded Operators, Lecture notes in Mathematics 1699"Springer. 241 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Kurose,Alfons Van Daele and Yinhuo Zhang: "Corepresentation theory of multiplier Hopf algebras II"International Journal of Mathematics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Kurose: "Quantum Doubles of Hopf *-algebras"RIMS Kokyuroku,Kyoto Univ.. 1110. 68-76 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Jean-Pierre Antoine,Atsushi Inoue and Hidekazu Ogi: "Standard generalized vectors and *-automorphism groups of partial O_*-algebras"Reports on Mathematical Physics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Atushi Inoue,Witold Karwowski and Hidekazu Ogi: "Standard weights on algebras of unbounded operators"Journal of Math.Soc.Japan. 51. 911-936 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Asao Arai: "Strong anticommutativity of Dirac operators on boson-fermion Fock spaces and representations of a supersymmetry algebra"Mathematische Nachrichten. 207. 61-77 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Nakazato: "Quintic elliptic curves arising in numerical range"Bulletin Sci.Tech.Hirosaki Univ.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Asao Arai: "Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials: Canonical commutation relations,quantum algebras" Journal of Mathematical Physics. 39. 2476-2498 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hidekazu Ogi: "Unbounded C^*-seminorms and *-algebras" RIMS Kokyuroku,Kyoto Univ.1077. 52-61 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Atsushi Inoue: "Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture Notes in Mathematics 1699" Springer, 241 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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