| 研究実績の概要 |
The first topic I studied during this year concerned quiver with potentials associated with triangulations of surfaces. In the paper "Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules", arXiv:1404.7269, join with Xueyu Luo, we study frozen Jacobian algebras coming from triangulations of polygons with one puncture. We relate it to the cluster category of type D_n which has been of a fundamental importance in the understanding of some cluster algebras.
The second project, in collaboration with Osamu Iyama and Gustavo Jasso concerned combinatorial properties of the exchange graph of so-called tau-tilting modules over finite dimensional algebras. In the paper "tau-rigid finite algebras and g-vectors", arXiv:1503.00285, we study the case where the number of tau-tilting modules is finite. We prove in this case that the corresponding simplicial complex is homeomorphic to a sphere. We also prove that the finiteness of tau-tilting modules is equivalent to the fact that every torsion class is functorially finite.
The third project, in collaboration with Osamu Iyama, concerned applications of orders to categorification of cluster algebras. In the paper "Lifting preprojective algebras to orders and categorifying partial flag varieties", arXiv:1503.02362, we relate categories of Cohen-Macauley modules over certain orders to categories of modules over certain finite dimensional factor algebras of these orders. We apply it to the categorification of cluster algebra structures of partial flag varieties, using preprojective algebras.
|
| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Compared to the initial project, though I did not progress so much about mutations at several vertices, I finished the project about triangulations of once punctures polygon with Xueyu Luo and I went much further on the topic of representations of orders related to categorification of cluster algebras with Osamu Iyama. Moreover, the project about tau-tilting modules can be inserted in the framework of the second year project concerning topology of exchange graph and g-vectors.
|
