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](https://arxiv.org/abs/math/0402054).


In [the file](http://www2.math.uni-paderborn.de/fileadmin/Mathematik/AG-Krause/ps/zelevinsky.ps), 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. 



What is the status of a problem about cluster categories?
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:


<https://arxiv.org/abs/1403.0549>


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 $4$.


