Cluster-tilting objects in higher cluster categories

doi:10.3868/s140-DDD-023-0017-x

Front. Math. China

2023, Vol. 18

Issue (3) : 187-201 https://doi.org/10.3868/s140-DDD-023-0017-x

RESEARCH　ARTICLE

Cluster-tilting objects in higher cluster categories

Xinhong CHEN¹(

), Ming LU²(

)

¹. Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
². Department of Mathematics, Sichuan University, Chengdu 610064, China

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Abstract

We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective, and also the properties for cluster-tilting objects in d-cluster categories. We get the following results: (1) When $d > 1$ , any almost complete cluster-tilting object in d-cluster category has only one complement. (2) Cluster-tilting objects in d-cluster categories are induced by tilting modules over some hereditary algebras. We also give a condition for a tilting module to induce a cluster-tilting object in a d-cluster category. (3) A 3-cluster category of finite type admits a cluster-tilting object if and only if its type is $A 1, A 3, D 2 n − 1 (n > 2)$ . (4) The $(2 m + 1)$ -cluster category of type $D 2 n − 1$ admits a cluster-tilting object such that its endomorphism algebra is self-injective, and its stable category is equivalent to the $(4 m + 2)$ -cluster category of type $A 4 m n − 4 m + 2 n − 1$ .

Keywords Almost complete cluster-tilting object Calabi−Yau triangulated category cluster-tilting object complement d-cluster category

Corresponding Author(s): Xinhong CHEN,Ming LU

Online First Date: 16 November 2023 Issue Date: 07 December 2023

Cite this article:

Xinhong CHEN,Ming LU. Cluster-tilting objects in higher cluster categories[J]. Front. Math. China, 2023, 18(3): 187-201.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-023-0017-x
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I3/187

Fig.1 AR-arrow diagram of

C 3 (A 3)

Fig.2 AR-arrow diagram of

C 3 (A n)

Fig.3 AR-arrow diagram of

C 2 d + 1 (A n)

Fig.4 AR-arrow diagram of

C 2 m + 1 (D 2 n ? 1)

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