2014 Fiscal Year Research-status Report
| Project/Area Number |
26800008
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| Research Institution | Nagoya University |
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Principal Investigator |
DEMONET Laurent 名古屋大学, 多元数理科学研究科, その他 (70646124)
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | quiver with potential / Jacobian algebras / Cohen-Macaulay modules / tau-tilting theory / cluster tilting theory / exchange graphs / cluster algebras / partial flag varieties |
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| Outline of Annual Research Achievements |
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.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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.
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| Strategy for Future Research Activity |
I have several plans for the coming year. I plan to extend the definition of frozen Jacobian algebras to any triangulation of any surface with boundaries and punctures and to investigate their order structures. I also expect to obtain nice results linking Brauer graph algebras and Jacobian algebras using these methods.
I also plan to study further the case of tau-rigid infinite algebras from the point of view of the combinatorics of their exchange graph. I will also continue to think about mutation at several vertices.
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| Causes of Carryover |
Small left over
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| Expenditure Plan for Carryover Budget |
Add to travel expenses
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