Integrable models and quantum cluster algebras

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540203
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: KUNIBA ATSUO 東京大学, 総合文化研究科, 教授 (70211886)
• Co-Investigator(Renkei-kenkyūsha): YAMADA Yasuhiko 神戸大学, 理学部, 教授 (00202383) ; NAKANISHI Tomoki 名古屋大学, 多元数理研究科, 教授 (80227842) ; OKADO Masato 大阪市立大学, 理学研究科, 教授 (70221843)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 可積分系 / 量子群 / ヤン・バクスター方程式 / 四面体方程式 / 量子座標環 / 3次元反射方程式 / PBW基底 / 表現論

Outline of Final Research Achievements

Several results illuminating novel mathematical structures of 3-dimensional integrable systems are obtained. (1) By invoking the representation theory of quantized coordinate ring of SP(4), two kinds of solutions are constructed for the 3-dimensional reflection equation proposed by Isaev and Kulish in 1997 for the first time. Its combinatorial limit, the classical analogue and a polynomial formula are obtained. (2) The transition coefficients of the PBW bases of the positive part of the quantized universal enveloping algebra Uq(g) are shown to coincide with the matrix elements of the intertwiner of the Soibelman representations of the quantized coordinate ring of g for all g of finite classical type. (3) For the two solutions known as R-operator and L-operator of the tetrahedron equation, 2-dimensional reduction is performed, and the results are identified with the quantum R matrices for generalized quantum groups including quantum superalgebras.

Free Research Field

数理物理学