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Representations of Groups, Lie Algebras, and Algebras

Research Project

Project/Area Number 04452004
Research Category

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

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionOsaka University

Principal Investigator

KAWANAKA Noriaki  OSAKA UNIV., DEPT.MATHEMATICS PROFESSOR, 理学部, 教授 (10028219)

Co-Investigator(Kenkyū-buntansha) SATAKE Ikuo  OSAKA UNIV., DEPT.MATHEMATICS INSTRUCTOR, 理学部, 助手 (80243161)
YAMANE Hiroyuki  OSAKA UNIV., DEPT.MATHEMATICS INSTRUCTOR, 理学部, 助手 (10230517)
MURAKAMI Jun  OSAKA UNIV., DEPT.MATHEMATICS ASSOCIATE PROFESSOR, 理学部, 助教授 (90157751)
NAGATOMO Kiyokazu  OSAKA UNIV., DEPT.MATHEMATICS ASSOCIATE PROFESSOR, 理学部, 助教授 (90172543)
MIYANISHI Masayoshi  OSAKA UNIV., DEPT.MATHEMATICS PROFESSOR, 理学部, 教授 (80025311)
平地 健吾  大阪大学, 理学部, 助手 (60218790)
新田 貴士  大阪大学, 理学部, 助手 (20202244)
小林 毅  大阪大学, 理学部, 講師 (00186751)
Project Period (FY) 1992 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥5,500,000 (Direct Cost: ¥5,500,000)
Fiscal Year 1993: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1992: ¥3,000,000 (Direct Cost: ¥3,000,000)
KeywordsErnst equation / Knot invariant / Hecke algebra / Quantum group / Singularity / Super Lie algebra / 特異点 / 超リー代数 / ホップ代数 / 局所ワイル群 / デル・ペッツオ曲面 / 基本群 / エルンスト方程式 / ヘーガード分解 / オービホールド / ベルグマン核 / テンパリー・リーブ代数
Research Abstract

Nagatomo constructed infinitely many rational solutions of the Ernst equation in general relativity. Murakami constructed a vertex type state model in Turaev's sense for the multi-variable Alexander polynomial. By using this model, a new set of axioms for the multi-variable Alexander polynomial is obtained. Murakami also determined the structure of the centralizer algebra of the mixed tensor representations of the quantum group U_q(gl(n, c)). This algebra can be considered as a generalization of the lwahori Hecke algebra of type A.Using this algebra, Murakami generalized the Yamada polynomial, which is an invariant of embeddings of a spatial graph in S^3. Murakami also investigated representations of the category of tangles using Kontsevich's interated integral, and succeeded in giving a combinatorial description of Kontsevich's integrals of knots, links and tangles. Satake investigated the simple elliptic singularity of type E6^^^, and constructed explicitly the flat theta invariants in the sense of K.Saito. Yamane constructed a new quasi-triangular Hopf algebra by quantizing the enveloping algebras of simple Lie superalgebras. He defined this Hopt algebra by generators and relations similarly as Drinfeld and Jimbo did for simple Lie algebras. His main contribution is the descovery of new kinds of defining relations which have no connterpart in the case of simple Lie algebras. Even for the enveloping algebras of simple Lie superalgebra, these relations were not known in this precise form. Yamane's results will have applications in mathematical physics and low dimensional topology. For example, he gave a formula for the universal R-matrix of his quasi-triangular Hopt algebra, and using this formula he found a representation theoretic interpretation for the R-matrix of Perk and Schultz.

Report

(3 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] K.Nagatomo: "Rational solutions of the Ernst equation" Advanced Studies in Pure Mathematics. 22. 189-196 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] J.Murakami: "A state model for the multi-variable Alexander polynomial" Pacific Journal of Mathematics. 157. 109-135 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] J.Murakami: "The Yamada polynomial of spatial graphs and knit algebras" Communications in Mathematical Physics. 155. 511-522 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Kosuda and J.Murakami: "Centralizer algebras of the mixed tensor representations of the quantum group Uq(gl(n,C))" Osaka Journal of Mathematics. 30. 475-507 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] I.Satake: "Flatstructure for the simple elliptic singularity of type E_6 and Jacobi form" Proceedings of the Japan Academy,Series A. 69. 247-251 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Yamane: "Quantized enveloping algebras associated to simple Lie superalgebras and their universal R-matrices" Publications of the Reserch Institute for Mathematical Sciences. 30(近刊). (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] K.Nagatomo: "Rational solutions of the Ernst equation" Advanced Studies in Pure Mathematics. 22. 189-196 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] J.Murakami: "A state model for the multi-variable Alexander polyomial" Pacific Journal of Mathematics. 157. 109-135 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] J.Murakami: "The Yamada polynomial of spatial graphs and knit algebras" Communications in Mathematical physics. 155. 511-522 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Kosuda and J.Murakami: "Centralizer algebras of the mixed tensor representations of the quantum group U_q(gl(n, c))" Osaka Journal of Mathematics. 30. 475-507 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] I.Satake: "Flat structure for the simple elliptic singularity of type E6^^^ and Jacobi form" Proceedings of the Japan Academy, Series A.69. 247-251 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Yamane: "Quantized enveloping algebras associated to simple Lie superalgebras and their universal R-matrices" Publications of the Reserch Institute for Mathematical Sciences. 30, (to appear). (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] J.Murakami: "A state model for the multi-variable Alexander polynomial" Pacific Journal of Mathematics. 157. 109-135 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] J.Murakami: "The Yamada polynomial of spatial graphs and knit algebras" Communications in Mathematical Physics. 155. 511-522 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] M.Kosuda and J.Murakami: "Centralizer algebras of the mixed tensor representations of the quantum group Uq(gl(n,C))" Osaka Journal of Mathematics. 30. 475-507 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] I.Satake: "Flat structure for the simple elliptic singularity of type E_6 and Jacobi form" Proceedings of the Japan Academy,Series A. 69. 247-251 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] H.Yamane: "Quantized enveloping algebras associated to simple Lie superalgebras and their universal R-matrices" Publications of the Reserch Institute for Mathematical Sciences. 30(近刊). (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] C.U.Hang,M.Miyanishi,K.Nishida,D.Q.Zhang: "On algebras which resemble the local Weyl algebra" Osaka Journal of Mathematics. 29. 393-404 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] M.Miyanishi,D.Q.Zhang: "Gorenstein log del Pezzo surfaces 2" Journal of Algebra. 155. (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] K.Nagatomo: "Rational solutions of the Ernst equations" Advanced Studies in Pure Mathematics. 22.

    • Related Report
      1992 Annual Research Report
  • [Publications] T.Kobayashi: "A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings has pdynomial growth" Osaka Journal of Mathematics. 29. 653-674 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] T.Nitta,K.Galicki: "Non-zero scalar curvature generalizations of the ALE hyperkahler metrics" Journal of Mathematical Physics. 33. 1765-1771

    • Related Report
      1992 Annual Research Report
  • [Publications] K.Hirachi: "The second variation of the Bergman kernel of ellipsoids" Osaka Journal of Mathematics.

    • Related Report
      1992 Annual Research Report

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Published: 1992-04-01   Modified: 2016-04-21  

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