Fischer groups based on vertex operator algebras

• Project/Area Number: 21740011
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tokyo Woman's Christian University (2010-2011); Aichi University of Education (2009)
• Principal Investigator: YAMAUCHI Hiroshi 東京女子大学, 現代教養学部, 講師 (40452213)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: フィッシャー群 / 頂点作用素代数 / 散在型有限単純群 / ヴィラソロ代数 / 3互換群 / モンスター単純群 / 有限幾何

Research Abstract

We study the vertex operator algebra VF which affords a natural action of the largest Fischer group of degree 24.We discovered a correspondence between the transpositions of the Fischer group and c=6/7 Virasoro vertex operator subalgebras of VF. Based on this correspondence, we obtained a framework which enables us to explain the 3 transposition property of the Fischer group and a mysterious relation between the largest Fischer group and the extended E6 Dynkin diagram. We also study the vertex operator superalgebra VB which affords a natural action of the Baby-monster sporadic simple group and found a one-to-one correspondence between the transpositions of the Baby-monster and c=7/10 Virasoro vertex operator subalgebras of VB. The 4-transposition property of the Baby-monster and its relation to the extended E7 Dynkin diagram is also explained via our results. We derived trace formulae for the extended Griess algebras which generalize the Matsu-Norton trace formulae and we found its fruitful applications to the sporadic finite simple groups.

Research Products

• [Journal Article] McKay's E6 observation on the largest Fischer group (2012)
  • Author(s): Gerald Hoehn, Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: Communications in Mathematical Physics Volume: Vol. 2 Issue: 2 Pages: 329-365
  • DOI: 10.1007/s00220-011-1413-8
  • Peer Reviewed
• [Journal Article] McKay's E7 observation on the Babymonster (2012)
  • Author(s): Gerald Hoehn, Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: International Mathematics Research Notices Volume: Vol. 2012 Issue: 1 Pages: 166-212
  • DOI: 10.1093/imrn/rnr009
  • Peer Reviewed
• [Journal Article] McKay's E7 observation on the Baby Monster (2011)
  • Author(s): Ching Hung Lam, Gerald Hohn, Hiroshi Yamauchi
  • Journal Title: International Mathematics Research Notices Volume: rnr009 Pages: 1-47
  • Peer Reviewed
• [Journal Article] The FLM conjecture and framed VOA, Vertex operator algebras and related areas (2009)
  • Author(s): Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: Contemporary Mathematics Pages: 125-138
  • Peer Reviewed
• [Journal Article] The FLM conjecture and framed VOA (2009)
  • Author(s): Ching Hung Lam, Hiroshi Yamauchi
  • Journal Title: Contemporary Mathematics 497 Pages: 125-138
  • Peer Reviewed
• [Presentation] 拡大グライス代数と松尾-ノートンの跡公式 (2012)
  • Author(s): 山内博
  • Organizer: 有限群とその表現、頂点作用素代数、組合せ論の研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-03-05
• [Presentation] 3A代数と3互換群,有限群と頂点作用素代数 (2012)
  • Author(s): 山内博
  • Place of Presentation: 東京女子大学
  • Year and Date: 2012-02-17
• [Presentation] 3A代数と3互換群 (2012)
  • Author(s): 山内博
  • Organizer: 有限群と頂点作用素代数
  • Place of Presentation: 東京女子大学
  • Year and Date: 2012-02-17
• [Presentation] Matsuo-Norton trace formula for vertex operator superalgebras, Conference on Vertex Operator Algebras (2011)
  • Author(s): 山内博
  • Organizer: Finite Groups and Related Topics
  • Place of Presentation: Institute of Mathematics, Academia Sinica(台湾)
  • Year and Date: 2011-12-19
• [Presentation] Matsuo-Norton trace formula for vertex operator superalgebras (2011)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Conference on Vertex Operator Algebras, Finite Groups and Related Topics
  • Place of Presentation: Institute of Mathematics, Academia Sinica, Taipei, Taiwan(招待講演)
  • Year and Date: 2011-12-19
• [Presentation] Trace identity and axial vectors for the Baby-monster, Conformal Field Theory (2011)
  • Author(s): 山内博
  • Organizer: Automorphic Forms and Related Topics
  • Place of Presentation: The MAThematics Center Heidelberg, Heidelberg University(ドイツ)
  • Year and Date: 2011-09-19
• [Presentation] Trace identity and axial vectors for the Baby-monster (2011)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Conformal Field Theory, Automorphic Forms and Related Topics
  • Place of Presentation: The MAThematics Center Heidelberg, Heidelberg University, Germany(招待講演)
  • Year and Date: 2011-09-19
• [Presentation] 3互換群に関連する代数 (2011)
  • Author(s): 山内博
  • Organizer: 第23回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2011-07-31
• [Presentation] 枠付き頂点作用素代数入門 (2010)
  • Author(s): 山内博
  • Organizer: 第54回Encounter with Mathematics
  • Place of Presentation: 中央大学
  • Year and Date: 2010-10-01
• [Presentation] 有限群と軌道体理論 (2010)
  • Author(s): 山内博
  • Organizer: 第22回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2010-07-30
• [Presentation] 散在型有限単純群と頂点作用素代数 (2010)
  • Author(s): 山内博
  • Organizer: 日本数学会2010年度年会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-27
• [Presentation] n-transposition groups and vertex operator algebras (2009)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Lie algebras, vertex algebras and automorphic forms
  • Place of Presentation: 15 South College Street, Edinburgh
  • Year and Date: 2009-09-04
• [Presentation] Admissible representation of affine VOAs and modular invariance (2009)
  • Author(s): 山内博
  • Organizer: Algebras, Groups and Geometries 2009
  • Place of Presentation: 玉原国際セミナーハウス
  • Year and Date: 2009-08-20
• [Presentation] 互換群と巾等元 (2009)
  • Author(s): 山内博
  • Organizer: 第21回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2009-08-03
• [Presentation] n-transposition groups and vertex operator algebras (2009)
  • Author(s): 山内博
  • Organizer: 第12回代数群と量子群の表現論研究集会
  • Place of Presentation: 休暇村南紀勝浦
  • Year and Date: 2009-05-14
• [Presentation] n-transposition groups and vertex operator algebras, Lie algebras, vertex algebras and automorphic forms (2009)
  • Author(s): 山内博
  • Place of Presentation: South College Street, Edinburgh(スコットランド)
• [Remarks] ホームページ(プレプリントを公開中)
• [Remarks]
  • URL: http://www.springerlink.com/content/17n025n467n73660/
• [Remarks]
  • URL: http://imrn.oxfordjournals.org/content/early/2011/02/20/imrn.rnr009.abstract
• [Remarks]
  • URL: http://arxiv.org/abs/1002.1777