Weight modules and crystal bases for quantum symmetric pairs

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14286
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Osaka Metropolitan University (2022-2023); Osaka City University (2021); Kyoto University (2020)
• Principal Investigator: Watanabe Hideya 大阪公立大学, 数学研究所, 学振特別研究員(PD) (10848782)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 量子対称対 / 表現論 / 結晶基底 / 標準基底

Outline of Final Research Achievements

The basic of the theory of crystal bases for quantum symmetric pairs, which describes combinatorially the representation theory of quantum symmetric pairs, was established. This theory enables one to understand various complex representations via elementary calculation. Also, applications to the representation theory of Lie algebras and discrete dynamics were discovered.
We also investigated the canonical bases of quantum symmetric pairs, which play a central role in the structure theory of quantum symmetric pairs, with an idea of crystal bases in mind. As a result, it turned out that the canonical bases are stable.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

本研究において、量子対称対の結晶基底の理論の基礎が初めて築かれ、また結晶基底の安定性が確認された。これらの結果は、当該研究分野(量子対称対の表現論)だけでなく、リー代数の表現論、量子群の表現論、離散力学系、可積分系、幾何学的表現論、圏論的表現論など多数の分野にまたがるものであり、様々な方向にさらなる発展を促すものである。