Application of cluster algebras to difference equations and 3-manifolds

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400037
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Chiba University
• Principal Investigator: Yamazaki Rei (井上玲) 千葉大学, 大学院理学研究院, 准教授 (30431901)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 数理物理学 / クラスター代数 / 可積分系 / 結び目不変量 / 幾何クリスタル

Outline of Final Research Achievements

We have studied knot invariants and integrable difference equations by applying cluster algebras. We realized the braid group using cluster mutations on a punctured disk, and clarified the relation with Kashaev's R-matrix via quantum cluster algebra. We studied the symplectic structure for the difference equations associated with exchange matrices of period one. We introduced a generalization of the discrete Toda lattice equation by using the network model on a torus, and solved its initial value problem using algebraic geometry and combinatorics. Further, we constructed the symmetric group action on a quiver on a cylinder, and studied the geometric R-matrix of A-type from the view point of cluster algebra.

Free Research Field

数理物理学