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Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras

Research Project

Project/Area Number 12440003
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

KITAZUME Masaaki  Chiba University, Faculty of Science, Professor, 理学部, 教授 (60204898)

Co-Investigator(Kenkyū-buntansha) SUGIYAMA Ken-ichi  Chiba University, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
NOZAWA Sohei  Chiba University, Faculty of Science, Professor, 理学部, 教授 (20092083)
KOSHITANI Shigeo  Chiba University, Faculty of Science, Professor, 理学部, 教授 (30125926)
HARADA Masaaki  Yamagata University, Faculty of Mathematical Sciences, Associate Professor, 理学部, 助教授 (90292408)
YAMADA Niromichi  Hitotsubashi University, Graduate School of Economics, Professor, 経済学研究科, 教授 (50134888)
宗政 昭弘  九州大学, 数理学研究科, 助教授 (50219862)
宮本 雅彦  筑波大学, 数学系, 教授 (30125356)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥6,500,000 (Direct Cost: ¥6,500,000)
Fiscal Year 2002: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥2,200,000 (Direct Cost: ¥2,200,000)
KeywordsFinite Group / Simple Group / Sporadic Simple Group / Monster Simple Group / Code / Lattice / Design / Vertex Operator Algebra / モンスター純群 / W代数 / グラフ
Research Abstract

We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :
1.The classifications of the 2-and 3-radical subgroups of Fischer's simple groups F_<22>, F_<23>, F'_<24> have been completed.
2.Even unimodular Gaussian lattices of dimension 12 have been classified.
3.A sufficient condition for extremal Z_6-codes, and as an application, many Z_6-code related the Leech lattice had been constructed. Moreover the self-dual Z_6-codes of length 8 are classified.
4.The construction method of unimodular lattices using ternary self-dual codes has been studied. Extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time.
5.The constructions of vertex operator algebras using Z_8-codes and subalgebras V_<√<2>A_3> have been considered. We also give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_3)^8 into the Leech lattice.
6.By using Z_3-orbifold construction given by Dong-Mason, we give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_2)^<12> the Leech lattice. We also give the explicit actions of certain 3A-elements of the Monster simple group on V^* are defined.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] M.Harada, M.Kitazume: "Z_4-Code Constructions for the Niemeier Lattices and Their Embeddings in the Leech Lattice"European Journal of Combinatorics. 21. 473-485 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, K.Koike, H.Shiga: "Notes on a certain integral orthogonal group and its quotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. 54. 1-15 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto, H.Yamada: "Borwein Identity and Vertex Operator Algebras"Journal of Number Theory. 82. 100-108 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Decomposition of the Moonshine vertex operator algebra as Virasoro modules"Journal of Algebra. 226. 893-919 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "A class of vertex operator algebras constructed from Z_8 codes"Journal of Algebra. 242. 338-359 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto: "3-transposition automorphism groups of VOA"Advanced Studies in Pure Mathematics. 32. 315-324 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, A.Munemasa: "Even Unimodular Gaussian Lattices of Rank 12"Journal of Number Theory. 95. 77-94 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Harada, M.Kitazume: "Z_6-Code Constructions of the Leech Lattice and the Niemeier Lattices"European Journal of Combinatorics. 23. 573-581 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2,0) 【cross product】 L(7/10,0) 【cross product】 L(4/5,0) 【cross product】 L(1,0)-modules"Journal of Pure and Applied Algebra. 173. 15-48 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, S.Yoshiara: "The radical subgroups of the Fischer simple groups"Journal of Algebra. 255. 22-58 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Harada, M.Kitazume, M.Ozeki: "Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over Z_6"Journal of Algebraic Combinatorics. 16. 209-223 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Harada, M.Kitazume: "Z_4-Code Constructions for the Niemeier Lattices and Their Embeddings in the Leech Lattice"European Journal of Combinatorics. 21. 473-485 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, K.Koike, H.Shiga: "Notes on a certain integral orthogonal group and its quotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. 54. 1-15 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto, H.Yamada: "Borwein Identity and Vertex Operator Algebras"Journal of Number Theory. 82, No.1. 100-108 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Decomposition of the Moonshine vertex operator algebra as Virasoro modules"Journal of Algebra. 226. 893-919 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "A class of vertex operator algebras constructed from Z_8 codes"Journal of Algebra. 242. 338-359 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto: "3-transposition automorphism groups of VOA"Advanced Studies in Pure Mathematics. 32. 315-324 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, A.Munemasa: "Even Unimodular Gaussian Lattices of Rank 12"Journal of Number Theory. 95. 77-94 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Harada, M.Kitazume: "Z_6-Code Constructions of the Leech Lattice and the Niemeier Lattices"European Journal of Combinatorics. 23. 573-581 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2, 0) 【cross product】 L(7/10, 0) 【cross product】 L(4/5, 0) 【cross product】 L(1, 0)-modules"Journal of Pure and Applied Algebra. 173. 15-48 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, S.Yoshiara: "The radical subgroups of the Fischer simple groups"Journal of Algebra. 255. 22-58 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Harada, M.Kitazume, M.Ozeki: "Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over Z_6"Journal of Algebraic Combinatorics. 16. 209-223 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kitazume, A.Munemasa: "Even Unimodular Gaussian Lattices of Rank 12"Journal of Number Theory. 95. 77-94 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kitazume, M.Harada: "Z_6-Code Constructions of the Leech Lattice and the Niemeier Lattices"European Journal of Combinatorics. 23. 573-581 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2,0)【cross product】(7/1O,0)【cross product】(4/5,0)【cross product】(1,0)-modules"Journal of Pure and Applied Algebra. 173. 15-48 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kitazume, S.Yoshiara: "The radical subgroups of the Fischer simple groups"Journal of Algebra. 255. 22-58 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Harada, M.Kitazume, M.Ozeki: "Ternary Code Construction of Unimodular Lattices"Journal of Algebraic Combinatorics. 16. 209-223 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "3-state Potts model, Moonshine vertex operator algebra and 3A-elements of the Monster group"International Mathematical Research Notices. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kitazume, M.Miyamoto: "3-transposition automorphism groups of VOA"Advanced Studies in Pure Mathematics. 32. 315-324 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "A class of vertex operator algebras constructed from Z_8 codes"Journal of Algebra. 242. 338-359 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "Moonshine Vertex Operator Algebra as L(1/2,0)【cross product】L(7/10,0)【cross product】L(4/5,0)【cross product】L(1,0)-modules"Journal of Pure and Applied Algebra. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kitazume, M.Harada: "Z_6-Code Constructions of the Leech Lattice and the Niemeier Lattices"European Journal of Combinatorics. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kitazume, A.Munemasa: "Even Unimodular Gaussian Lattices of Rank 12"Journal of Number Theory. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kitazume,M.Miyamoto: "3-transposition automorphism groups of VOA"Advanced Studies in Pure Mathematics. (発表予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kitazume,C.H.Lam,H.Yamada: "A class of vertex operator algebras constructed from Z_8 codes"Journal of Algebra. (発表予定). (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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