Finding a cluster-tilting object for a representation finite cluster-tilted algebra

Abstract

We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case. The distribution of cluster-tilting objects of type D nIn this section we will show how to, given the quiver Q of a cluster-tilted algebra of type D n , explicitly find a cluster-tilting object in the cluster category of type D n inducing it. The goal is to prove the following theorem:Theorem 4.1. Given the quiver Q of a cluster-tilted algebra of type D, it will be of one of the types 1) ⋆ ⋆ 2) ⋆ ⋆ 3)

Periodicity of cluster tilting objects

“…• if a vertex has three neighbours then two of the adjacent arrows belong to a 3-cycle and the last arrow adjacent to the vertex does not belong to any 3-cycle. The next result is not explicitly stated in [4], but follows directly from results in [4]. It shows that the position of a vertex in the quiver of a cluster tilted algebra is closely related to which τ -orbit the corresponding indecomposable summand lies in, in the AR-quiver of the cluster catgory.…”

“…The first classification of quivers of cluster tilted algebras of type D n was given in [9] also by Vatne and contained four main types. In [4], Oppermann, Bertani-Økland and Wrålsen were able to reduce this description to three main types of quivers. We will refer to the classification as it is presented in [4].…”

“…In [4], Oppermann, Bertani-Økland and Wrålsen were able to reduce this description to three main types of quivers. We will refer to the classification as it is presented in [4].…”

Special Biserial Cluster-Tilted Algebras

“…From Lemma 6 it is clear which quivers correspond to special biserial cluster-tilted algebras. Let Q T be a cycle, then it is known by [4,Theorem 4.1] that all the indecomposable summands of T are -objects, and that if Y is one indecomposable summand of T then the other summands of T are given as follows:…”

“…In [4] this was combined into three types. We will use the latter description, and we state the details needed to discuss which quivers in the mutation class correspond to special biserial cluster-tilted algebras.…”