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.

通过 Deboret，我们对所有温柔代数的挠率类（可能是无限维，也可能是全局维无限）进行分类。我们的分类是在紧凑的真实二维表面上采用简单的曲线和叠层；这项工作在其他领域可能很有用，例如拓扑深谷范畴，其中温和代数至关重要。通过 Adachi，我们研究了布劳尔图代数的复形。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倾斜复形。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，我们研究了前簇倾斜理论和最小Auslander-Gorenstein的推广。特别是，我们的结果统一了之前与 Marczinzik 在特殊二列对称代数研究方面的工作。与 Miemietz 一起，我们研究了法定 2-类别的 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重新整理联系起来，并发现将域上的余代数定位理论适当地推广到在法定 2 范畴中出现的余代数对象；该理论适用于其他领域的兴趣设置，例如幺半群范畴的模范畴的研究。我们与 Wong 一起研究了对称群上模表示设置中的置换模的 p 复形。我们研究了这些复合物的斜线同源性；特别是，我们证明了 Wildon 猜想的扩展。

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

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