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A new quality theorem for tensor category and the equivarence of topological quantum field theories

Research Project

Project/Area Number 10640019
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionGraduate School of Mathematics, Nagoya University

Principal Investigator

HAYASHI Takahiro  Nagoya University, Graduate School of Mathematics, assistant professor, 大学院・多元数理科学研究科, 助教授 (60208618)

Co-Investigator(Kenkyū-buntansha) OKADA Soichi  Nagoya University, Graduate School of Mathematics, Assistant professor, 大学院・多元数理科学研究科, 助教授 (20224016)
NAKANISHI Tomoki  Nagoya University, Graduate School of Mathematics, Assistant professor, 大学院・多元数理科学研究科, 助教授 (80227842)
TSUCHIYA Akihiro  Nagoya University, Graduate School of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (90022673)
OHTA Hiroshi  Nagoya University, Graduate School of Mathematics, Assistant professor, 大学院・多元数理科学研究科, 助教授 (50223839)
Project Period (FY) 1998 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1998: ¥900,000 (Direct Cost: ¥900,000)
Keywordsquantum group / tensor category / Tannaka duality / classical invariant theory / inverse matrix / Littlewood-Richardson 則 / Littlewood-Richardson則 / 6j-symbol / 位相的場の量子論
Research Abstract

For each finite split semisimple tensor category C, the canonical Tannaka duality gives a quantum group (face algebra) whose comodule category is equivalent to C. The duality gives a unified understanding of tensor categories arising from mathematics and physics, and also, it gives a new picture of the representation theory of the ordinary groups. By applying the duality and other techniques in the quantum group theory, we obtained the following results.
1. We showed that the quantum ej-symbo, is a sum of the partition functions of the ABF mode, of finite size. Also we gave a simple summation formula for the ordinary 6j-symbols.
2. By using the duality (or rather the canonical fiber functor), we constructed a bases for invariants and semiinvariants of binary quadratics and binary cubics.
3. For each tensor products of two irreducible representations of the general linear group, we gave their explicit irreducible decomposition at the module level.
4. We classified the braiding and th'e ribbon structure on each quantum classica, groups and the tensor category of type A of level L.
5. For each linearly reductive matrix group G, we give an "economical" inverse matrix formula for each elements of G.

Report

(5 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] T.Hayashi: "Face algebras and unitarity of SU(N)_L-TQFT"Communications in Mathematical Physics. 203. 211-247 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Hayashi: "Galois quantum groups of II_1-Subfactors"Tohoku Mathematical Journal. 51. 365-389 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"Journal of Algebra. 233. 614-641 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Hayashi: "A brief introduction to face algebras"Contemporary Mathematics. 267. 161-176 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Hayashi: "Face algebras and unitarity of su(n)-TOFT"communications in Mathematical Physics. 203. 211-247 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Hayashi: "Galois quantum groups of II-subfactors"Tohoku Mathematical Journal. 51. 365-389 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"Journal of Algebra. 233. 614-641 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Hayashi: "A brief introduction to face algebras"Contemporary Mathematics. 267. 161-176 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Hayashi: "Face algebras and unitarity of SU(N)_L-TQFT"Communications in Mathematical Physics. 203. 211-247 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hayashi: "Galois quantum groups of II_1-subfactors"Tohoku Mathematical Journal. 51. 365-389 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hayashi: "Coribbon Hopf(face) algebras generated by lattice modds"Journal of Algebra. 233. 614-641 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hayashi: "A brief introduction to face algebras"Contemporary Mathematics. 267. 161-176 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takahiro Hayashi: "Face algebras and unitarity of SU(N)c-TQFT"Communications in Mathematical Physics. 203. 211-247 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takahiro Hayashi: "Galois quantum groups of II_1-subfactors"Tohoku Mathematical Journal . 51. 365-389 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takahiro Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"Journal of Algebra. 233. 614-641 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takahiro Hayashi: "A brief introduction to face algebras"Contemporary mathematics. 267. 161-176 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takahiro Hayashi: "Face algebras and Unitarity of SU(N)_L-TQFT"Communications in Mathematical Physics. 203. 211-247 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takahiro Hayashi: "Galois quantum groups of II_1 subfactors"Tohoku Mathematical Journal. 51. 365-389 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takahiro Hayashi: "Face algebras I - A generalization of quantum group theory" Journal of Mathematical society of Japan. vol50 No.2. 294-315 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Takahiro Hayashi: "Quantum groups and quantum semigroups" Journal of Algebra. 204. 225-254 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Takahiro Hayashi: Communications in Mathematical Physics.

    • Related Report
      1998 Annual Research Report
  • [Publications] Soichi Okada: "Applications of Minor Summation Formulas to Rectangular-shaped Representations of classical groups" Journal of Algebra. 205. 337-367 (1998)

    • Related Report
      1998 Annual Research Report

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

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