2019 Fiscal Year Research-status Report
Cluster theory through derived categories and self-injective algebras
| Project/Area Number |
18K03238
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| Research Institution | Nagoya University |
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Principal Investigator |
ダルポ エリック 名古屋大学, 多元数理科学研究科(国際), 准教授 (00785959)
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| Project Period (FY) |
2018-04-01 – 2021-03-31
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| Keywords | Derived category / Cluster category / Cluster tilting / Silting |
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| Outline of Annual Research Achievements |
The research during this year has proceeded in two main directions. The first direction is the investigation of n-cluster-tilting in derived categories of algebras of finite global dimension, using the concept of silting. Here, the notion of an orbital n-cluster-tilting subcategory has been generalised using silting, giving a tool to understand a large class of n-cluster-tilting subcategories of derived categories in a unified framework. The question arises which n-cluster-tilting subcategories of derived categories arise as such generalised orbital subcategories.
Related to this, some work was also done to clarify the connection between local boundedness in the module category respectively the stable module categories of a locally bounded k-linear category.
The second direction is the investigation of functors between cluster categories of Dynkin type that was initiated in 2018. Here, some preliminary results have been obtained in the direction of extending the existing results for Dynkin type A to Dynkin type D. The main idea has been to transport results from type A to type D, by writing algebras of type D as skew-group algebras of algebras of type A, and using the resulting adjunction between the derived categories of the respective algebras to carry over the results.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The work has been slightly delayed due to a higher-than-expected teaching load during this year.
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| Strategy for Future Research Activity |
The primary goal of the coming year is to continue the two directions of investigation pursued in 2019: functors between higher cluster categories of Dynkin type, and n-cluster-tilting in derived categories.
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| Causes of Carryover |
The cost for international travel was somewhat lower than expected. Also, a planned invitation of a foreign visitor was postponed.
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