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.

利用Demonet，我们对温和代数（可能是无限维数和可能是无限维数）的所有扭转类进行了分类。我们的分类是在紧致的实二维表面上使用简单的曲线和分层；这样的工作在其他领域也有潜在的用处，比如拓扑深谷范畴，其中温和代数是至关重要的。利用Adachi研究了Brauer图代数的复形。我们采用类似于Khovanov-Seidel结果的拓扑技术来构造这些代数的预拟复合体。我们还计算了它们的自同态环，并研究了它们倾斜的可能性。与Iyama和Marczinzik一起，我们研究了预簇倾斜理论的推广和极小Auslander-Gorenstein。特别地，我们的结果统一了先前与Marczinzik关于特殊双列属对称代数的研究。利用Miemietz，我们研究了任意2类的2表示的短精确序列的概念。我们将这些概念与阿贝尔范畴的溯及性联系起来，发现域上的协代数的局部化理论可以推广到任意2范畴中的协代数对象；这一理论也适用于其他领域的兴趣设置，如一元范畴的模范畴的研究。利用Wong，研究了对称群上模表示设置下置换模的p-复形。我们研究了这些配合物的斜线同源性；特别地，我们证明了Wildon猜想的一个扩展。

项目成果

Aaron Chan @ Nagoya

陈富城@名古屋

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

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

Generalising hereditary algebras and so on

推广遗传代数等

• 发表时间: 2018
• 作者: Adachi Takahide;Aihara Takuma;Chan Aaron;Aaron Chan;Aaron Chan
• 通讯作者: Aaron Chan

Curves on marked surfaces and complexes of Brauer graph algebras

标记曲面上的曲线和布劳尔图代数的复形

• 发表时间: 2018
• 作者: Adachi Takahide;Aihara Takuma;Chan Aaron;Aaron Chan;Aaron Chan;Aaron Chan
• 通讯作者: Aaron Chan