2017 Fiscal Year Annual Research Report
曲面の組合せ論によるブラウアーグラフ代数の導来圏の研究
| Project/Area Number |
17F17019
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| Research Institution | Nagoya University |
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Principal Investigator |
伊山 修 名古屋大学, 多元数理科学研究科, 教授 (70347532)
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| Co-Investigator(Kenkyū-buntansha) |
CHAN AARON 名古屋大学, 多元数理科学研究科, 外国人特別研究員
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| Project Period (FY) |
2017-04-26 – 2019-03-31
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| Keywords | homological algebra / surface topology / special biserial algebra / quasi-hereditary algebra / Koszul duality / Cohen-Macaulay module / Iwanaga-Gorenstein ring / Auslander-Reiten theory |
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| Outline of Annual Research Achievements |
During the first year of the JSPS International Fellowship, I have started new projects with existing as well as new collaborators, apart from joint works with the host. The work in progress are as follows. With Laurent Demonet, we are in progress on classifying all torsion classes of Brauer graph algebras, as well as their idempotent quotients which include gentle algebras; the classification we use is to employ simple curves and laminations on compact oriented real-two-dimensional surfaces; such work is potentially useful in other areas such as topological Fukaya categories where gentle algebras are of central importance. With Takahide Adachi, we continue our study on tilting theory of Brauer graph algebras, started in a recently accepted joint paper with Takuma Aihara. With Takahide Adachi and Mayu Tsukamoto, we studied Auslander-Dlab-Ringel algebras which are strongly quasi-hereditary and standard Koszul; we showed that such an algebra also possesses other strong homological properties that are often of interest in Lie theoretic representation theory; the article is in preparation.
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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
We can already formulate the classification of torsion classes in terms of combinatorics of topological objects on surfaces, what remains is to check if there are any results of a similar form in topology and write down a rigourous argument for our guess.
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| Strategy for Future Research Activity |
We will continue to study tilting theory surrounding algebras that are of close relation to Brauer graph algebras, and if the progress is good, we will go on to study how the classification of torsion classes could allow us to understand t-structures or silting complexes of these algebras.
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