Cluster-tilting objects in higher cluster categories

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_{2n-1}\left(n>2\right). (4) The (2m+1)-cluster category of type D_{2n-1} admits a cluster-tilting object such that its endomorphism algebra is self-injective, and its stable category is equivalent to the (4m+2)-cluster category of type A_{4mn-4m+2n-1}.

Graphical abstract

Keywords

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

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Higher Education Press 2023