2017 Fiscal Year Research-status Report
Mutation in derived categories and lattice theory of torsion classes and wide subcategories
| Project/Area Number |
17K14160
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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(国際), G30特任准教授 (70646124)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | Representation theory / Torsion classes / Lattice theory / Brauer graph algebras |
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| Outline of Annual Research Achievements |
This year, I have been finishing and submitting the article in collaboration with Osamu Iyama, Nathan Reading, Idun Reiten and Hugh Thomas about the lattice theory of torsion classes. It turned out that we got better results than we expected as most results we get about torsion classes are true even when there are infinitely many torsion classes. In particular, we extend the notion of congruence uniform lattice to the case of complete infinite lattices. This result is surprising both from an algebraic point of view and from a lattice theoretical point of view, as it shows that tors A has rather unique properties, compared to classical and well studied lattices. In particular, we show that its Hasse quiver contains a big part of its information, even though it is an infinite lattice.
In parallel, I started with Aaron Chan a project of classification of torsion classes over Brauer graph algebras. We expect them to be classified by systems of non-crossing, unbounded curves on some compact surfaces up to homotopy. In this way, it extends in a natural way the classical notion of laminations, extensively studied by William Thurston. Moreover, we have in mind a purely combinatorial description of the lattice structure of these torsion classes.
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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
The plans for 2017 have been achieved correctly, with even certain improvements.
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| Strategy for Future Research Activity |
I plan to continue this year the project, going more deeply into the case with infinitely many torsion classes, and the relation with scattering diagrams in particular.
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