2015 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 |
This year, I finished the project "Lifting preprojective algebras to orders and categorifying partial flag varieties" with Osamu Iyama. We succeeded to improve the results in order to have a categorification for any Dynkin type. The first report about the paper has been positive.
I also introduced and studied "algebras of partial triangulations" ( arXiv:1602.01592). We introduce two classes of algebras coming from partial triangulations of marked surfaces. The first one, called frozen, is generally of infinite rank and contains frozen Jacobian algebras of triangulations of marked surfaces. The second one, called non-frozen, is always of (explicit) finite rank and contains non-frozen Jacobian algebras of triangulations of marked surfaces and Brauer graph algebras. We classify the partial triangulations, the frozen algebras of which are lattices over a formal power series ring. For non-frozen algebras, we prove that they are symmetric when the surface has no boundary. From a more representation theoretical point of view, we prove that these non-frozen algebras of partial triangulations are at most tame and we define a combinatorial operation on partial triangulation, generalizing Kauer moves of Brauer graphs and flips of triangulations, which give derived equivalences of the corresponding non-frozen 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
My project, even if it is taking sometimes different directions is still progressing. Meanwhile, I have some other results which are very related to the main direction.
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| Strategy for Future Research Activity |
I have some common project in progress with Osamu Iyama and Gustavo Jasso about the combinatorics of mutations and some projects with Osamu Iyama to continue the categorification project. I also need to improve the paper about algebras of partial triangulations, and I plan to study more deeply their representation theory.
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| Causes of Carryover |
Some travel plans (conferences and invitations) have been delayed.
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| Expenditure Plan for Carryover Budget |
I will probably have more conference to attend this year.
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