| 研究実績の概要 |
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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