研究者詳細 - CHAN Aaron Kay Yam

論文 - CHAN Aaron Kay Yam

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On representation-finite gendo-symmetric algebras with only one non-injective projective module 査読有り

Aihara Takuma, Chan Aaron, Honma Takahiro

JOURNAL OF ALGEBRA 603 巻頁： 61 - 88 2022年8月

記述言語：英語掲載種別：研究論文（学術雑誌）出版者・発行元：Journal of Algebra

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig [15,16] addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have precisely one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi [21]. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.

DOI： 10.1016/j.jalgebra.2022.04.002

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Scopus

COIDEMPOTENT SUBCOALGEBRAS AND SHORT EXACT SEQUENCES OF FINITARY 2-REPRESENTATIONS 査読有り国際共著

Chan Aaron, Miemietz Vanessa

NAGOYA MATHEMATICAL JOURNAL 243 巻頁： 316 - 341 2021年9月

記述言語：日本語掲載種別：研究論文（学術雑誌）出版者・発行元：Nagoya Mathematical Journal

In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian 2-representations.

DOI： 10.1017/nmj.2020.1

Web of Science

Scopus

Irreducible representations of the symmetric groups from slash homologies of p-complexes

Aaron Chan

Algebraic Combinatorics 2021年2月

On simple-minded systems and tau-periodic modules of self-injective algebras

Chan Aaron, Liu Yuming, Zhang Zhen

JOURNAL OF ALGEBRA 560 巻頁： 416 - 441 2020年10月

記述言語：日本語掲載種別：研究論文（学術雑誌）出版者・発行元：Journal of Algebra

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, C a stably quasi-serial component (i.e. its stable part is a tube) of rank n of the Auslander-Reiten quiver of A, and S be a simple-minded system of the stable module category mod_A. We show that the intersection S∩C is of size strictly less than n, and consists only of modules with quasi-length strictly less than n. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of A cannot be in any simple-minded system.

DOI： 10.1016/j.jalgebra.2020.05.024

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Scopus

Auslander-Gorenstein algebras from Serre-formal algebras via replication 査読有り

Chan Aaron, Iyama Osamu, Marczinzik Rene

ADVANCES IN MATHEMATICS 345 巻頁： 222 - 262 2019年3月

記述言語：日本語掲載種別：研究論文（学術雑誌）出版者・発行元：Advances in Mathematics

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga–Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras – called the replicated algebras – given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander–Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander–Gorenstein algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi–Yau. We apply these results to a construction of algebras from Yamagata [29], called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.

DOI： 10.1016/j.aim.2019.01.010

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Scopus

On Representation-Finite Gendo-Symmetric Biserial Algebras 査読有り

Chan Aaron, Marczinzik Rene

ALGEBRAS AND REPRESENTATION THEORY 22 巻 ( 1 ) 頁： 141 - 176 2019年2月

記述言語：日本語掲載種別：研究論文（学術雑誌）出版者・発行元：Algebras and Representation Theory

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

DOI： 10.1007/s10468-017-9760-6

Web of Science

Scopus

Diagrams and discrete extensions for finitary 2-representations 査読有り

Chan, A., Mazorchuk, V.

Mathematical Proceedings of the Cambridge Philosophical Society 166 巻 ( 2 ) 頁： 325 - 352 2019年

Classification of two-term tilting complexes over Brauer graph algebras 査読有り

Adachi Takahide, Aihara Takuma, Chan Aaron

MATHEMATISCHE ZEITSCHRIFT 290 巻 ( 1-2 ) 頁： 1 - 36 2018年10月

記述言語：日本語掲載種別：研究論文（学術雑誌）出版者・発行元：Mathematische Zeitschrift

Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.

DOI： 10.1007/s00209-017-2006-9

Web of Science

Scopus

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

Aaron CHAN, Steffen KOENIG, Yuming LIU

Journal of Pure and Applied Algebra 219 巻 ( 6 ) 頁： 1940-1961 2014年8月

Two-term tilting complexes and simple-minded systems of self-injective Nakayama algebras

Aaron CHAN

Algebra and Representation theory 18 巻 ( 1 ) 頁： 183-203 2014年6月

Some homological properties of tensor and wreath products of quasi-hereditary algebras

Aaron CHAN

Communication in Algebras 42 巻 ( 6 ) 頁： 2368-2379 2014年2月

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