Quantum toroidal algebras and quantum integrable systems

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03549
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Rikkyo University
• Principal Investigator: JIMBO Michio 立教大学, 名誉教授, 名誉教授 (80109082)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: quantum toroidal algebra / deformed W algebra / integrals of motion

Outline of Final Research Achievements

We studied quantum toroidal algebras in view of its applications to integrable systems. We obtained the following results: 1)We determined the branching rule of Wakimoto representations of quantum toroidal gl_n to its subalgebra which is a product of gl_1 quantum toroidal algebras. This shows explicitly that the deformation of the coset W algebra of type gl_n/gl_{n-1} is given by deformed W superalgebra W(gl_{n|n-1}). 2) We introduced an algebra K_1, which is a comodule over gl_1 quantum toroidal algebra, gave a uniform description of deformed W algebras of classical types, and constructed a commutative subalgebra (integrals of motion) thereof. 3) We generalized the algebra K_1 to a gl_n analog K_n, and constructed its commutative subalgebra.

Free Research Field

可積分系

Academic Significance and Societal Importance of the Research Achievements

W代数は共形場理論の数学的定式化である。量子トロイダル代数はW代数のq変形の研究に有力な方法を与えている。これまでの研究は概ねA型の場合に限られていたが、本研究では量子トロイダル代数を少し拡張することによって、一般の場合の統一的扱いに一歩を踏み出した。特に「localな運動の保存量」と呼ばれる可換な部分代数の構成がA型の場合とほぼ同じ方法でできることがわかり、今後の研究への素材を提供できたと考えている。