前射影多元環の導来圏の研究

原投影代数的派生范畴的研究

• 批准号: 17J00652
• 负责人: 水野 有哉
• 金额: $ 3.99万
• 依托单位: Shizuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2017
• 资助国家: 日本
• 起止时间: 2017-04-26 至 2020-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17J00652/
• 关键词: 箙の表現論; 傾理論; 前射影多元環; ブラウアー樹木多元環; τ傾理論; 導来圏; 加群圏; 導来同値; 変異; 傾複体; クイバー

今年度の主要な成果として次の３つをあげる事が出来る.(I) Affine型前射影多元環の傾複体の研究. 導来圏同値を理解する基本的手法は,傾複体を理解する事である.以前までの自身の研究において,Dynkinグラフに付随する前射影多元環の場合には,その傾複体を完全に分類する事に成功している.この研究の目的はそれをaffineグラフに付随する前射影多元環の場合に拡張する事である.共同研究においてaffine型前射影多元環の長さが２の傾複体を調べ,τ傾理論を用いる事でその変異をaffine Weyl群と対応させ,完全な分類を与える事に成功した.さらに任意の傾複体は環をシフトさせたものから変異で得られると予想し,実際に特別な場合に成り立つ事を示すに至った.(II) 前射影多元環,道多元環の表現論とコクセター群の統括的研究. 80年頃P.Gabrielにより示されたDynkin型道多元環の表現論と正ルートとの対応は,現在ではクイバーの表現論とLie理論とのもっとも基本的な結びつきの一つと言える.近年になり,より直接的な結びつきが前射影多元環の表現論とCoxeter群へと拡張した形で得られる事が分かってきた.共同研究において最新の前射影多元環の理論を用いて,道多元環の表現論とCoxeter群との根本的な関連性を考察し,道多元環上のねじれ対をCoxeter群の言葉によって明確に記述出来る事を示した.(III) 単体的複体を用いたBrauer樹木多元環の傾理論の研究. Brauer樹木多元環はmodular表現論から現れる極めて重要なクラスである.共同研究においてBrauer樹木多元環の長さ２の傾複体を,単体的複体(多面体)として捉え全体の数がBrauer樹木多元環の頂点の数にのみ依存する事を示した.そして結果としてこの数は導来不変量になる事を明らかにした.

The main achievements of this year have been published for three times. (I) Affine-type pre-projective multi-environmental complex research. We have been led to understand the basic techniques and complex ways to understand things. In the past, our own research has been successful, and Dynkin has been successful in projecting multiple environments in the past. The purpose of the study is to make sure that the affine environment is projected in a multi-environmental environment before the project. The joint study of the multi-projective environment in front of the environmental protection system is based on the fact that the affine Weyl group is divided into different categories and successfully classified. If you want to use any complex environment, please do not think much about it, and the international community will be responsible for the development of the project. (II) Projective multiplicity and multivariate environmental tables discuss the comprehensive study of the environmental protection group. In the year 80's, the P.Gabriel survey showed that the multi-environmental table of the Dynkin model was in full swing. Now we are going to discuss the Lie theory, the basic results of the review, and the basic information. In recent years, there has been a direct projection of multiple environments in the past few years, which shows that the Coxetern group has received significant changes in the shape of the environment. Joint research on the latest multi-projective environmental theory of environmental theory, multi-dimensional environmental theory shows the fundamental relationship between coxtimulars, Coxeter group speech on the multi-component environment, and the events are clearly documented. (III) the complex of the system is studied by means of the Brauer multi-environmental theory. The Brauer multivariate environment modular table discusses the importance of important environmental issues. Joint research on the Brauer multivariate environment complex, the complex (polyhedron) of the complex (polyhedron), the total number of Brauer, the number of points of the multivariate environment, the dependence of the environment, the number of the complex (polyhedron), the number of the complex (polyhedron) of the complex (polyhedron), the number of points of the multivariate environment, the number of points of the multivariate environment, the number of points, the number of nodes, the number of points, and the dependence of the environment. The results show that there is a lot of information and information about the number of people.

项目成果

Torsion pairs for quivers and the Coxeter groups

箭袋和 Coxeter 组的扭转副

• 发表时间: 2018
• 作者: Takahide Adachi;Yuya Mizuno;水野 有哉;水野 有哉;水野 有哉
• 通讯作者: 水野 有哉

Sortable elements and torsion pairs for quivers

箭袋的可排序元件和扭转副

• 发表时间: 2019
• 期刊: Proceedings of the 51st Symposium Ring theory and Representation theory
• 作者: Mizuno Yuya;Mizuno Yuya
• 通讯作者: Mizuno Yuya

Torsion pairs for quivers and the Weyl group

箭袋和韦尔群的扭转副

• 发表时间: 2018
• 作者: S. Koshitani;J. Mueller;F. Noeske;I. Mori;Tomoyuki Arakawa;Tomoyuki Arakawa;Hiroshi Isozaki;Osamu Iyama;Izuru Mori and Kenta Ueyama;Tomoyuki Arakawa;水野有哉
• 通讯作者: 水野有哉

Two-sided tilting complexes over preprojective algebras

原射代数上的两侧倾斜复形

• 发表时间: 2017
• 作者: Hiroyuki Minamoto;Kota Yamaura;越谷 重夫;Tomoyuki Arakawa;S. Koshitani;Tomoyuki Arakawa;S. Koshitani;Tomoyuki Arakawa;Masahisa Sato;水野有哉;Tomoyuki Arakawa;水野有哉
• 通讯作者: 水野有哉

Derived Picard Groups of Preprojective Algebras of Dynkin Type

Dynkin型原投影代数的派生Picard群

• DOI: 10.1093/imrn/rny299
• 发表时间: 2019
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Mizuno Yuya