Cluster categories and cluster-tilted algebras

Abstract

We have given an introduction to the theory of cluster categories and cluster-tilted algebras, and this was one of our main objectives in this thesis. We have seen that cluster-tilted algebras are relation-extension algebras, and this gave us a way of constructing the quiver of a cluster-tilted algebra from a tilted algebra. A cluster-tilted algebra of finite representation type is determined by its quiver, and this raised questions about the generality of this result. We defined a new class of algebras, namely cluster-relation algebras, and saw that the Cartan determinant of a cluster-relation algebra in some cases can be found directly from its quiver. Cluster-tilted algebras of finite representation type are cluster-relation algebras. We asked questions about when a cluster-relation algebra is cluster-tilted, and which cluster-tilted algebras are cluster-relation algebras. Another interesting problem mentioned, is to classify which quivers can occur as cluster-tilted algebras. The java-application JMutation has been developed, and it is described in Appendix A. We used this application to compute the number of cluster-tilted algebras of type $A_n$ and $D_n$ for small $n$. It could be possible to find functions $a(n)$ and $d(n)$, giving the number of cluster-tilted algebras of type $A_n$ and $D_n$ respectively. As we can see, there are many open problems waiting to be answered.