Investigation of similarities in representation theory of quantum affine algebras of several different Dynkin types

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14515
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Tokyo Institute of Technology (2022); Shibaura Institute of Technology (2019-2021)
• Principal Investigator: Oya Hironori 東京工業大学, 理学院, 准教授 (90835505)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 量子ループ代数 / 有限次元表現論 / 量子Grothendieck環 / q-指標 / クラスター代数

Outline of Final Research Achievements

In my joint work with Ryo Fujita, David Hernandez, and Se-jin Oh, we constructed a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories of finite-dimensional modules over the quantum loop algebras associated with complex simple Lie algebras g_1 and g_2 under the assumption that the complex simple Lie algebra defined as the unfolding of g_1 coincides with that of g_2. As its application, we deduced several new positivity properties of the simple (q,t)-characters of non-symmetric types, non-trivial birational relations among the simple (q,t)-characters of quantum loop algebras associated with g_1 and g_2, and the affirmative answer to Hernandez's conjecture for the quantum loop algebra of type B_n(1).

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

非対称型量子ループ代数の有限次元表現圏の量子Grothendieck環の構造の研究を，対称型量子ループ代数の有限次元表現圏の量子Grothendieck環と比較するという新しいアブローチから進めた．結果として，適切に対応する型のものを比較すると(q,t)-指標を保つ良い同型が存在することが示され，これまで非対称型の場合に知られていなかった(q,t)-指標の正値性や，B_n(1)型の量子ループ代数の既約表現のq-指標が代数的なアルゴリズムで計算されることが証明された．さらに，非対称型と対称型の量子ループ代数の有限次元表現の間の新たな関係を示唆する，(q,t)-指標の間の関係を導くことができた．