Representation theory of quantum deformed current algebras

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17565
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Shinshu University
• Principal Investigator: Wada Kentaro 信州大学, 学術研究院理学系, 准教授 (60583862)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 表現論 / 量子群 / リー代数 / q-Schur 代数 / Hecke 代数

Outline of Final Research Achievements

We studied on some structures and representations of quantum deformed current algebras.
We obtained the following results on structures of quantum deformed current algebras. The quantum current algebra is a Hopf subalgebra of a quantum loop algebra, and the quantum deformed current algebra is a coideal subalgebra of the quantum current algebra. These imply that the module category of the quantum deformed current algebra is a module category over the module category of the quantum current algebra which is a tensor category. We also found that the quantum deformed current algebra is a subalgebra of a shifted quantum affine algebra introduced by Finkelberg-Tsymbaliuk.
We also classify the isomorphism classes of finite dimensional simple modules of quantum deformed current algebras.

Free Research Field

数学(代数学, 表現論)

Academic Significance and Societal Importance of the Research Achievements

量子変形カレント代数は, 巡回 q-Schur 代数の表現論を動機として, 研究代表者によって導入された代数であるが, 今回の研究によって, その "量子群" としての構造が明らかになったことによって, 巡回 q-Schur 代数の表現論や関連する表現論も含め, 今後多くの応用が期待できる。また, 数理物理に現れるクーロン枝の数学的な定式化に関連して導入された, シフト量子アフィン代数との関係も発見できたことは重要な成果である。