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Representation theory of infinite-dimensional Lie algebras and superalgebras and its mathematical applications

Research Project

Project/Area Number 10440009
Research Category

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

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

WAKIMOTO Minoru  Faculty of Mathematics, Prof., 大学院・数理学研究院, 教授 (00028218)

Co-Investigator(Kenkyū-buntansha) SATO Eiichi  Faculty of Mathematics, Prof., 大学院・数理学研究院, 教授 (10112278)
TAGAWA Hiroyuki  Wakayama Unive., Faculty of Education, Ass. Prof., 教育学部, 助教授 (80283943)
YAMADA Mieko  Kanazawa Univ., Faculty of Science, Prof., 理学部, 教授 (70130226)
KAGEI Yoshiyuki  Faculty of Mathematics, Ass. Prof., 大学院・数理学研究院, 助教授 (80243913)
鈴木 昌和  九州大学, 大学院・数理学研究科, 教授 (20112302)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2000: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1999: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsaffine superalgebra / integrable representation / Appell's elliptic function / asymptotics / character formula / W-algebra / N=2 superconformal algebra / admissible representation / super-conformal代数 / Drinfeld-Sokalov reduction / BRST複体 / conformal-super代数 / super conformal代数 / 指標 / modular変換 / テータ函数 / 楕円函数 / criticalレベル / super-Boson-fermion対応 / fusion係数 / toroidalリー環 / ソリトン方程式
Research Abstract

Under this Grant-in-Aid, I made joint research with Professor Victor G.Kac, and obtained the following results :
1. "Integrable representations" for affine superalgebras are never easy concept, but should be treated carefully. We found that integrable representations consist of two kinds, namely principal-integrable representations and subprincipal-integrable representations, and gave the explicit and complete list of all highest weights for principal and subprincipal integrable modules.
2. We gave an explicit construction of fundamental sl(m|n)^- and osp(m|n)^-modules by using free bosons and free fermions. Using this explicit construction, we calculated the characters and obtained three kinds of character formulas --- Weyl-Kac type, theta-function type and quasi-particle type. From these character formulas, we found that the characters of fundamental sl(m|1)^-modules are Appell's elliptic functions which were discovered by Appell in 1880's but have been forgotten over one hundred years … More . These functions are not modular functions, but we succeeded to compute their asymptotics.
3. The trivial representation of an affine superalgebra sl(2|2)^ is a representation of critical level, since its dual Coxeter is equal to 0. So there was no known denominator identity for such superalgebras. We obtained explicitly the denominator formula for sl(2|2)^ by using Riemann's theta relations.
4. It is known by the theory of Frenkel-Kac-Wakimoto (1994) that the W-algebra of an usual affine Lie algebra and its representations are constructed in terms of the quantized Drinfeld-Sokolov reduction. But, for affine superalgebras, an immediate extension of this method fails to give a right W-algebra, and the construction of the W-algebra associated to affine superalgebras has long been a problem. We succeeded to resolve the difficulty by tensoring the factor, which arises from the algebraic variety, with the usual BRST-complex. The W-algebra of an affine superalgebra sl (2|1)^ obtained by this method is the direct sum of the centerless Virasoro algebra and the N=2 superconformal algebra. This theory enables us to make a detail investigation on representations of the N=2 superconformal algebra by means of admissible representations of sl(2|1)^. Actually we found that, other than the usual minimal series representations, there exist curious series of N=2 representations whose characters are half-modular functions. This research is now in progress very intensively. Less

Report

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

    (34 results)

All Other

All Publications (34 results)

  • [Publications] Minoru Wakimoto: "Representation theory of affine superalgebras at the critical level"Documenta Mathematica Extra Volume II ICM 1998. 605-614 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kenji Iohara et al: "Hirota bilinear forms with 2-toroidal symmetry"Phys.Letters. A254. 37-46 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kenji Iohara et al: "Notes on differential equations arising from a representation of 2-toroidal lie algebras"Progr.Theoret.Phys.Suppl.. 135. 166-181 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Minoru Wakimoto: "Modular transformation of admissible representations and fusion algebras associated to non-symmetric transformation matrices"Advanced Studies in Pure Math.. 26. 325-353 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shun-Jen Cheng et al: "Extensions of Neveu-Schwarz conformal modules"Journal of Math.Physics. 41. 2271-2294 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Victor G.Kac et al: "Integrable highest weight modules over affine superalgebras and Appell's function"Commun.Math.Phys. 215. 631-682 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 脇本実: "無限次元Lie環"岩波書店. 343 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Minoru Wakimoto: "Infinite-Dimensional Lie Algebras"American Mathematical Society. 304 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Wakimoto: "Representation theory of affine superalgebras at the critical level"Documenta Mathematica, Proceedings of ICM. Vol.II. 605-614 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Wakimoto: "A construction of N=2 and centerless N=4 superconformal fields via affine superalgebras"Nuclear Physics. B530. 665-682 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S-J.Cheng, V.G.Kac, M.Wakimoto: "Extensions of conformal modules"in "Topological Field Theory, Primitive Forms and Related Topics" , Birkhauser. 665-682 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Iohara, Y.Saito, M.Wakimoto: "Hirota bilinear forms with 2-toroidal symmetry"Phys.Letters. A254. 37-46 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Iohara, Y.Saito, M.Wakimoto: "Notes on differential equations arising from a representation of 2-toroidal Lie algebras"Prog.Theoret.Phys.Suppl.. 135. 166-181 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Wakimoto: "Modular transformation of twisted characters of admissible representations and fusion algebras associated to non-symmetric transformation matrices"Advanced Studies in Pure Math. 26. 325-353 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S-J.Cheng, V.G.Kac, M.Wakimoto: "Extensions of Neveu-Schwarz conformal modules"Jour. of Math.Phys.. 41. 2271-2294 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] V.G.Kac and M.Wakimoto: "Integrable highest weight modules over affine superalgebras and Appell's function"Commun.Math.Phys.. 215. 631-682 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Wakimoto: "Infinite-Dimensional Lie Algebras (in Japanese)"Iwanami Shoten Publishers. 343 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Wakimoto: "Infinite-Dimensional Lie Algebras, (Translations of Mathematical Monographs, 195, Iwanami Series in Modern Mathematics)"American Mathematical Society. 304 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Minoru Wakimoto: "Modular transformation of admissible representations and fusion algebras associated to non-symmetric transformation matrices"Advanced Studies in Pure Math.. 26. 325-353 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shun-Jen Cheng et al.: "Extensions of Neveu-Schwarz conformal modules"Journal of Math.Physics. 41. 2271-2294 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Victor G.Kac et al.: "Integrabl highest weight modules over affine superalgebras and Appell's functions"Commun.Math.Phys.. (to appear)(in press). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Koichi Shinoda et al: "A family of Hadamard matrices of dihedral group type"Discrete Applied Math.. 102. 141-150 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroyuki Tagawa: "A recursion formula of the weighted parabolic Kazhdan-Lusztig polynomials"Advanced Studies in Pure Math.. 28. 373-389 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Eiichi Sato: "Behavior of rational curves with the minimal degree in projective space bundle in any characteristic"Jour.Kyoto.Univ.. (to appear)(in press). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Minoru Wakimoto: "Infinite-Dimensional Lire Algebras"American Mathematical Society. 304 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 庵原謙治: "Hirota bilinear forms with 2-toroidal symmetry"Physics Letters A. 254. 37-46 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 庵原謙治: "Notes on Differential Eqnations Arising from a Representation of 2-Toroidal Lie algebras"Progress of Theor. Phys. Supplement. 135. 166-181 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 脇本 実: "Modular transformation of twisted characters of admissible representations and fusion algebras associated to non-symmetric transformation matrices"Advanced Studies in Pure Math.. 26. (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 脇本 実: "無限次元Lie環(岩波講座 現代数学の展開3)"岩波書店. 343 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shun-Jen Cheng: "Extensions of conformal modules" Progress in Mathematics. 160. 79-129 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 脇本 実: "Representation theory of affine superalgebras at the critical level" Document a Mathematics Extra Volume. 2. 605-614 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 脇本 実: "A conctruction of N=2 and anterless N=4 super conformal feilds via affine superalegebras" Nuclear Physics B. 530. 665-682 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 脇本 実: "Modular transformation of twisted characters of admissible representations and fusion algelras assoiated to non-symmetric transformation matrius" Advanced Studies in Pure Math.26. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 庵原謙治: "Hirota bilinear forms with 2-teroidal symmetry" Physics Letters A. (1999)

    • Related Report
      1998 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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