Let $H$ be a hereditary algebra of Dynkin type. There is a cluster category $\mathcal{C}_H$ defined by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov in Tilting theory and cluster combinatorics.

In the file, Andrei Zelevinsky proposed the following problem.

Problem: find an explicit description of $End_H(T)$ ($T$ is a tilting object in $\mathcal{C}_H$) in terms of quivers and relations.

What is current status of this problem? Thank you very much.


For $H$ the path algebra of a 'star-shaped' quiver having three legs with lengths $r$, $s$, $t$, an answer seems to be implicit in Lamberti's combinatorial model for the cluster category:


but she doesn't make it explicit. This includes all Dynkin types, where $(r,s,t)\in\{(1,p,q),(2,2,n),(2,3,3),(2,3,4),(2,3,5)\}$. The answer for Dynkin types $\mathsf{A}$ and $\mathsf{D}$ is more explicit, because the quivers with relations are given by the Jacobian algebras associated to triangulations of a disk and a punctured disk respectively:

Caldero–Chapoton–Schiffler, type $\mathsf{A}$: https://arxiv.org/abs/math/0411238

Schiffler, type $\mathsf{D}$: https://arxiv.org/abs/math/0608264

This leaves just type $\mathsf{E}$, where one could in principle compute all of the (finitely-many!) cluster-tilted algebras using Lamberti's construction.

Some non-Dynkin hereditary algebras are also covered by triangulations of marked bordered surfaces, such as $\tilde{\mathsf{A}}$ by the annulus, but I am not sure precisely which ones. Most are not, since quivers arising from triangulations are very special; for example, the vertices have valency at most $4$.


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