Mutation in derived categories and lattice theory of torsion classes and wide subcategories

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K14160
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nagoya University
• 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)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: torsion class / lattice / gentle algebra / Representation theory / wide subcategory / Brauer graph algebra

Outline of Final Research Achievements

The class of gentle algebras A is a classical object in representation theory and is widely studied as a typical example of algebras of tame representation type. Recently it has been applied to cluster algebras and topological Fukaya categories. It is known that A-modules are classified by strings (walks on the quiver of A which are compatible with relations). Also A is realized as a dissected marked surface, and a string is realized as a curve on it. In particular, the notion of the intersections of two strings can be defined. In a joint work with A. Chan, we constructed a canonical one-to-one correspondence between the torsion classes of the category of finite dimensional A-modules and the maximal parametrized non-crossing sets of infinite strings. This is a powerful result that does not require the functorial finiteness of torsion classes.

Free Research Field

代数の表現論

Academic Significance and Societal Importance of the Research Achievements

加群圏のねじれ部分圏は、導来圏の特別なt構造の加群圏への制限であり、古典的な対象であるとともに、傾理論、特に変異理論の発展により、近年盛んに調べられている。本研究では、与えられた代数の加群圏におけるねじれ部分圏の全体の成す完備束(complete lattice)を調べ、その束論的性質や代数の表現論的性質との関係を明らかにした（Iyama, Reiten, Readin, Thomasとの共同研究）。また、gentle代数と呼ばれる重要な代数に対して、ねじれ部分圏を組み合わせ論的なデータによって完全に分類することに成功した（Chanとの共同研究）。