Study on induced modules of vertex operator algebras

• Project/Area Number: 15K04823
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Ehime University
• Principal Investigator: Abe Toshiyuki 愛媛大学, 教育学部, 教授 (30380215)
• Research Collaborator: LAM ching hung ; YMAMADA hiromichi
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 頂点作用素代数 / 表現論 / 誘導表現 / 普遍包絡環 / 量子次元 / 原田予想 / オービオフォールド / オービフォールド模型 / ツイスト加群

Outline of Final Research Achievements

On this study, we try to give a construction of induced modules of vertex operator algebras. We give a construction of universal enveloping algebras which give twisted modules and their generalizations. The construction of induced modules are usually based on the structure of module category with fusion product. But we use algebraic structure of vertex operator algebras. As related consequences we solve a conjecture on construction of moonshine vertex operator algebra and give a sufficient condition to hold Harada conjecture on finite group.

Academic Significance and Societal Importance of the Research Achievements

頂点作用素代数はムーンシャイン予想の解決を目的の一つとして構成された代数系である。可算無限個の演算を持つがその表現論はその導入以来多くの進展があった。その誘導表現の構成は圏論的構成が主流であるが、ここでは代数的構成を目指し、twisted 加群やその一般化を与える普遍包絡環を構成した。その過程でムーンシャイン頂点作用素代数の構成に関する予想を解決したり、有限群論の問題にも取り組むことができた。

Research Products

• [Int'l Joint Research] Academia Sinica(台湾)
• [Int'l Joint Research] Academia Sinica(台湾)
• [Journal Article] On Zp-orbifold constructions of the Moonshine vertex operator algebra (2018)
  • Author(s): Abe Toshiyuki、Lam Ching Hung、Yamada Hiromichi
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 1-2 Pages: 683-697
  • DOI: 10.1007/s00209-017-2036-3
  • Peer Reviewed / Int'l Joint Research
• [Presentation] 原田予想への一考察 (2018)
  • Author(s): 安部 利之
  • Organizer: 草津群論セミナー
• [Presentation] Extensions of tensor products of the VOA $V_{{\sqrt 2}A_n}^\sigma $ (2018)
  • Author(s): 安部 利之
  • Organizer: RIMS共同研究（公開型）「組合せ論的表現論の諸相」
• [Presentation] Extensions of tensor products of vertex operator algebras $V_{{\sqrt 2}A_n}^{\widehat{\sigma}} $ (2018)
  • Author(s): 安部 利之
  • Organizer: Vertex Operator Algebras and Related Topics
• [Presentation] Group-like fusion をもつ頂点作用素代数の構成について (2017)
  • Author(s): 安部利之
  • Organizer: 日本数学会2017年度年会
  • Place of Presentation: 首都大学東京（東京都・八王子市）
  • Year and Date: 2017-03-25
• [Presentation] A construction of VOAs having group-like fusion (2017)
  • Author(s): 安部利之
  • Organizer: Finite Groups and Vertex Operator Algebras 2017
  • Place of Presentation: 東京女子大学（東京都・杉並区）
  • Year and Date: 2017-03-21
  • Int'l Joint Research / Invited
• [Presentation] ムーンシャイン頂点作用素代数の Zp-軌道体構成について (2017)
  • Author(s): 安部利之
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] The Leech lattice VOA as an extension of V_{√2 A_4}^{\otimes 6} (2016)
  • Author(s): 安部利之
  • Organizer: 第３３回代数的組合せ論シンポジウム
  • Place of Presentation: 滋賀大学（滋賀県・大津市）
  • Year and Date: 2016-06-24
• [Presentation] Classification of irreducible modules of Z2×Z2-orbifold model of lattice vertex operator algebras (2015)
  • Author(s): 安部利之
  • Organizer: Vertex operator algebra and related topics
  • Place of Presentation: 成都（中国）
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited