New developments in the study of quantum groups

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03426
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Osaka Metropolitan University (2022); Osaka City University (2019-2021)
• Principal Investigator: OKADO Masato 大阪公立大学, 大学院理学研究科, 教授 (70221843)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 量子群 / 超リー代数 / 結晶基底

Outline of Final Research Achievements

Solutions of the reflection equation and the representation theory of quantum affine (hyper)algebras are studied. In the former, we first studied a method to obtain from solutions of the three-dimensional reflection equation, but gradually shifted to a method using the theory of iquantum groups. In the case of type A, we obtained explicit forms of the combinatorial K-matrices for all quasi-split Satake diagrams. In the latter, in the case of affine quantum groups associated with orthosymplectic Lie superalgebras, we defined a category of representations including q-oscillator representations and constructed a new family of representations. We also found that q-oscillator representations of type C and finite-dimensional representations of type D are mutually interpolated by an exact monoidal functor.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

反射方程式は、ヤン・バクスター方程式とともに境界付き１次元量子系や２次元統計力学系の可積分条件を与えるものである。この解が多く構成されたことによって可積分系研究への応用が期待される。量子アフィン超代数は、従来詳しく研究され、多くの応用をもたらした量子アフィン代数を拡張するものであり、超双対性を通じて量子アフィン代数の有限次元表現の圏とq-振動子表現の圏を関係づける。また、古くから表現論分野で知られているHoweの双対ペアの理論の量子アフィン類似とみなすこともでき、Howe理論を新しい視点で再構成することにも繋がる。