2020 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 – 2022-03-31
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| Keywords | fractionally Calabi-Yau / trivial extension / periodic |
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| Outline of Annual Research Achievements |
An important open problem in the homological algebra of self-injective algebras is to characterise periodic algebras. An algebra A is said to be periodic if it has a periodic projective resolution as an A-A-bimodule. In joint work, we have found a solution to this problem for trivial extension algebras: the trivial extension T(A) of a finite-dimensional algebra A is periodic if and only if A has finite global dimension and is fractionally Calabi-Yau. This result was proved by analysing the relative bar resolution of a certain differentially graded algebra quasi-isomorphic to T(A).
We also showed that so-called twisted periodicity of T(A) is equivalent to A being twisted fractionally Calabi-Yau, as well as to the d-representation-finiteness of the r-fold trivial extension algebra T_r(A) for some positive integers r and d.
Using these results, we have constructed a large number of new examples of periodic as well as fractionally Calabi-Yau algebras, and given answers to several open questions. A primary source of motivation for the above work is to understand the connection between periodicity properties (including the fractional Calabi-Yau property) and d-representation-finiteness in self-injective 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
The results obtained are substantial, and in line with the expectations.
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| Strategy for Future Research Activity |
We are currently working on a continuation of last year's research, studying d-cluster-tilting modules in trivial extension algebras T(A) of fractionally Calabi-Yau algebras A. In addition, if the time allows, I plan to study mutation theory of d-silting objects of derived categories, in particular how it relates to mutation of cluster-tilting subcategories.
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| Causes of Carryover |
Due to the ongoing pandemic, all travel and invitations of international visitors were cancelled last year. To the extent that the situation allows, such visits and invitations may be carried out during the fiscal year 2021 instead.
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