Geometry of semi-infinite flag varieties and level-zero representation theory of quantum affine algebras

• Project/Area Number: 26887002
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tokyo Denki University (2015); Tohoku University (2014)
• Principal Investigator: Ishii Motohiro 東京電機大学, 情報環境学部, 助教 (00732463)
• Project Period (FY): 2014-08-29 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 量子アフィン展開環 / 半無限Bruhatグラフ / 結晶基底 / 二重アフィンWeyl群 / アフィンLie環 / アフィンWeyl群 / 半無限旗多様体

Outline of Final Research Achievements

1. A gallery model for level-zero representations: We introduce a gallery model for crystal bases of level-zero extremal weight modules over quantum affine algebras.
2. Construction and classification of holomorphic VOA of central charge 24: We classify holomorphic VOAs of central charge 24 obtained by applying Miyamoto's $\mathbb{Z}_3$-orbifold construction.

Research Products

• [Journal Article] Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras (2016)
  • Author(s): Motohiro Ishii, Satoshi Naito and Daisuke Sagaki
  • Journal Title: Advances in Mathematics Volume: 290 Pages: 967-1009
  • DOI: 10.1016/j.aim.2015.11.037
  • Peer Reviewed
• [Journal Article] Automorphisms of Niemeier lattices for Miyamoto's Z_3-orbifold construction (2015)
  • Author(s): Motohiro Ishii, Daisuke Sagaki, and Hiroki Shimakura
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 1-2 Pages: 55-83
  • DOI: 10.1007/s00209-015-1413-z
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] 量子ループ代数の組合せ論的表現論 (2016)
  • Author(s): 石井基裕
  • Organizer: 数理科学談話会
  • Place of Presentation: 室蘭工業大学（北海道室蘭市）
  • Year and Date: 2016-01-20
  • Invited
• [Presentation] アフィングラスマン多様体入門 (2015)
  • Author(s): 石井基裕
  • Organizer: アフィングラスマン多様体とその周辺
  • Place of Presentation: 東京大学玉原国際セミナーハウス（群馬県沼田市）
  • Year and Date: 2015-08-18
  • Invited
• [Presentation] A combinatorial realization of Weyl modules over quantum affine algebras (2015)
  • Author(s): M. Ishii
  • Organizer: RIMS Camp-style seminar "Diagram algebras and topology"
  • Place of Presentation: 沖縄県糸満市
  • Year and Date: 2015-08-03
  • Int'l Joint Research / Invited
• [Presentation] 量子アフィン展開環のレベル・ゼロ表現に対するギャラリー模型 (2015)
  • Author(s): 石井基裕
  • Organizer: Reresentation Theory Seminar
  • Place of Presentation: 京都大学
  • Year and Date: 2015-01-09
  • Invited
• [Presentation] Path and gallery models for level-zero representations of quantum affine algebras (2014)
  • Author(s): 石井基裕
  • Organizer: RIMS研究集会「有限群とその表現、頂点作用素代数、代数的組合せ論の研究」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-12-17
  • Invited
• [Presentation] A gallery model for level-zero representations of quantum affine algebras (2014)
  • Author(s): Motohiro Ishii
  • Organizer: Shanghai Workshop on Representation Theory
  • Place of Presentation: Tongji University, China
  • Year and Date: 2014-12-08
  • Invited
• [Presentation] A gallery model for level-zero representations of quantum affine algebras (2014)
  • Author(s): 石井基裕
  • Organizer: RIMS研究集会「組合せ論的表現論と表現論的組合せ論」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-10-29