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2017 Fiscal Year Final Research Report

Studies on discrete integrable systems via tropical algebraic curves

Research Project

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Project/Area Number 26400107
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionChiba 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

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Published: 2019-03-29  

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