Study on infinite dimensional algebraic groups and Lie algebras, and application to quasi-periodic and aperiodic structures

• Project/Area Number: 17K05158
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Tsukuba
• Principal Investigator: Morita Jun 筑波大学, 数理物質系(名誉教授), 名誉教授 (20166416)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 代数群 / リー代数 / 代数的K理論 / 局所アフィン・リー代数 / カッツ・ムーディ群 / スキーム / 四元数体 / 量子ビット / カッツ・ムーデイ群 / Kac-Moody 群 / 基本同値 / 群スキーム / 結晶構造 / 準周期構造 / 非周期構造 / 構造論 / 表現論 / アフィン・リー代数 / 単純群 / 乗法因子群 / 局所アフィン・ルート系 / 表現

Outline of Final Research Achievements

(1)The structure of K_2SL_2(R) was determined for several prime numbers p_1,...,p_n, where R = [1/p_1,...,1/p_n]. (2) We classified minimal locally affine Lie algebras. This is a joint work with Yoji Yoshii. (3) We characterized affine Kac-Moody groups using schemes and Galois descert. This is a joint work with A. Pianzola and T. Shibata. (4) We discussed some infinite root system obtained from H4 root systems in quaternion division ring, and obtained a new application to quantum bits. This is a joint work with Robert Moody.

Academic Significance and Societal Importance of the Research Achievements

何れも有限次元および無限次元の代数群とリー代数に関わる基本的な研究成果である。新たな知見も多く含み、数学的な価値は高く、意義深いと認めとられる。

Research Products

• [Int'l Joint Research] バルイラン大学(イスラエル)
• [Int'l Joint Research] アルバータ大学(カナダ)
• [Int'l Joint Research] Poincare Institute(フランス)
• [Int'l Joint Research] Xiamen University(中国)
• [Int'l Joint Research] Calicornia Institute of Technology(米国)
• [Int'l Joint Research] University of Alberta(カナダ)
• [Journal Article] Classification of minimal locally affine Lie algebras (2023)
  • Author(s): Morita Jun、Yoshii Yoji
  • Journal Title: Journal of Algebra Volume: 616 Pages: 97-154
  • DOI: 10.1016/j.jalgebra.2022.11.004
  • Peer Reviewed / Open Access
• [Journal Article] Chevalley groups over Dedekind domains and some problems for K_2(2,Z_S) (2020)
  • Author(s): Jun Morita
  • Journal Title: Toyama Mathematical Journal Volume: 41 Pages: 83-122
  • Peer Reviewed
• [Journal Article] Simple Kac-Moody groups with trivial Schur multipliers (2018)
  • Author(s): Jun Morita
  • Journal Title: Science China Mathematics Volume: 61 Issue: 2 Pages: 311-316
  • DOI: 10.1007/s11425-016-9170-1
  • Peer Reviewed
• [Journal Article] Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers (2018)
  • Author(s): Robert Moody, Jun Morita
  • Journal Title: Communications in Algebras Volume: 46
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Chevalley groups over Dedekind domains and K2 groups (2020)
  • Author(s): Jun Morita
  • Organizer: The second meeting for Study of Number Theory, Hopf Algebras and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Kac-Moody algebras, groups and related topics (2018)
  • Author(s): 森田純
  • Organizer: Finite Groups, VOAs, and Related Topics 2018
  • Invited