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

利用表面组合学研究布劳尔图代数的派生范畴

• 批准号: 17F17019
• 负责人: 伊山 修
• 金额: $ 1.47万
• 依托单位: Nagoya 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 2019-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17F17019/
• 关键词: homological algebra; surface topology; special biserial algebra; quasi-hereditary algebra; Koszul duality; Cohen-Macaulay module; Iwanaga-Gorenstein ring; Auslander-Reiten theory

With Demonet, we classify all torsion classes of (possibly infinite dimensional and possibly global dimension infinite) gentle algebras. Our classification 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 Adachi, we study complexes of Brauer graph algebras. We employ topological techniques similar to results of Khovanov-Seidel to construct pretilting complexes of these algebras. We also calculate their endomorphism rings, and investigate the possiblity of them being tilting.With Iyama and Marczinzik, we study generalisation of precluster tilting theory and minimal Auslander-Gorenstein. In particular, our results unify a previous work with Marczinzik on the study of special biserial gendo-symmetric algebras.With Miemietz, we study the notion of short exact sequences for 2-representations of fiat 2-categories. We relate these notions with recollement of abelian categories, and found appropriate generalisation of localisation theory for coalgebras over a field to coalgebras objects arising in fiat 2-categories; this theory is applicable to setting of interests in other fields such as the study of module-category of a monoidal category.With Wong, we study p-complexes of permutation modules in the setting of modular representations over the symmetric groups. We investigate the slash homologies of these complexes; in particular, we prove an extension of a conjecture of Wildon.

使用Demonet，我们将所有扭转类别分类（可能是无限的维度和可能的全球尺寸无限）柔和的代数。我们的分类是在紧凑的实时二维表面上采用简单的曲线和层压。这种工作可能在其他领域（例如福卡亚类别）中有可能有用，而温和代数非常重要。与Adachi一起，我们研究了Brauer图代数的复合物。我们采用与Khovanov-Seidel的结果相似的拓扑技术来构建这些代数的替代复合物。我们还计算了它们的内态环，并研究了它们正在倾斜的可能性。与Iyama和Marczinzik一起，我们研究了预团体倾斜理论和最小Auslander-Gorenstein的概括。特别是，我们的结果统一了先前与Marczinzik在特殊的双性gendo-对称代数的研究中进行的一项工作。与Miemietz一起，我们研究了菲亚特2类别2类别的简短序列的概念。我们将这些概念与对阿贝尔类别的回忆联系起来，并发现对山地对山地的定位理论的适当概括是在菲亚特2类中引起的山结构的对象；该理论适用于在其他领域的兴趣设置，例如研究单体类别的模块类别的研究。与Wong一起，我们研究了在对称组的模块化表示的设置中，我们研究了置换模块的P复合。我们研究了这些复合物的斜线同源物。特别是，我们证明了一个猜想的延伸。

项目成果

Aaron Chan @ Nagoya

陈富城@名古屋

Diagrams and discrete extensions for finitary 2-representations

有限 2 表示的图表和离散扩展

• DOI: 10.1017/s0305004117000858
• 发表时间: 2019
• 期刊: Math. Proc. Cambridge Philos. Soc.
• 作者: Aaron Chan;Volodymyr Mazorchuk
• 通讯作者: Volodymyr Mazorchuk

Classification of two-term tilting complexes over Brauer graph algebras

• DOI: 10.1007/s00209-017-2006-9
• 发表时间: 2015-04
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: T. Adachi;T. Aihara;Aaron Chan

On Representation-Finite Gendo-Symmetric Biserial Algebras

• DOI: 10.1007/s10468-017-9760-6
• 发表时间: 2016-07
• 期刊: Algebras and Representation Theory
• 影响因子: 0.6
• 作者: Aaron Chan;René Marczinzik

On two construction of Iwanaga-Gorenstein algebras

Iwanaga-Gorenstein 代数的两种构造

• 发表时间: 2017
• 作者: Adachi Takahide;Aihara Takuma;Chan Aaron;Aaron Chan;Aaron Chan;Aaron Chan;Aaron Chan
• 通讯作者: Aaron Chan