Studies on discrete integrable systems via tropical algebraic curves

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400107
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Chiba University
• Principal Investigator: Nobe Atsushi 千葉大学, 教育学部, 准教授 (80397728)
• Co-Investigator(Kenkyū-buntansha): 間田 潤 日本大学, 生産工学部, 准教授 (80396853)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: トロピカル幾何学 / クラスター代数 / 戸田格子 / QRT系 / 楕円曲線

Outline of Final Research Achievements

First we considered Lie-algebraic generalizations of the Toda lattice. We realized the generalized Toda lattices of types A(2)2N, C(1)N and D(2)N as the sub-dynamical systems of the Toda lattices of types A(1)2N-1, A(1)2N, A(1)2N+1, respectively. We also obtained their tropical analogues.
Next we studied a tropical analogue of the Hessian group, which is the group of linear automorphisms acting on the Hesse pencil. We then obtained the dihedral group of degree 3 as the group of linear automorphisms acting on the tropical analogue of the Hesse pencil.
We moreover investigated the cluster algebras of rank 2 from the view point of discrete integrable systems. We gave the conserved quantities of the dynamical systems arising from the cluster algebras of types A1*A1, A2, B2, G2, A(1)1 and A(2)2. We also showed direct connections between the dynamical systems and the Mordell-Weil groups of the elliptic curves arising via the conserved quantities of the dynamical systems.

Free Research Field

大域解析学