Geometric realizations of cluster categories of non-simply-laced types

Question

Geometric realizations of cluster categories of simply-laced types are studied in the following papers.

1. Philippe Caldero, Frédéric Chapoton, and Ralf. Schiffler, Quivers with relations arising from clusters ($A_n$ case).

2. Ralf Schiffler, A geometric model for cluster categories of type $D_n$.

3. Lisa Lamberti, Combinatorial model for the cluster categories of type $E$.

I didn't find papers which studied geometric realizations of cluster categories of non-simply-laced types. Have this problem been studied? Thank you very much.

Answer 1

The main author I am aware of working on this problem is Demonet, see

1. Cluster algebras and preprojective algebras : the non simply-laced case
2. Categorification of skew-symmetrizable cluster algebras
3. Mutations of group species with potentials and their representations. Applications to cluster algebras

There is also an earlier paper of Dong Yang which I have not read:

4. Non-simply-laced Clusters of Finite Type via Frobenius Morphism

To my understanding, all of these papers use some form of folding construction to reduce to the skew-symmetric case. This is a problem, because there are quivers which cannot be unfolded! See Example 14.4 in

5. Strongly primitive species with potentials I: Mutations

This last paper attempts to provide a partial solution to the problem of categorifying the non-unfoldable case. You might also want to watch Daniel Labardini-Fragoso's talk from Zelevinsky's memorial conference; I thought it provided a very clear overview.