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On a quantum analog of the Caldero-Chapoton formula. (English) Zbl 1237.16013
Finding a closed formula for all cluster variables in a cluster algebra in terms of an initial cluster is a nontrivial task. A construction, due to Caldero and Chapoton, relates cluster variables to the representations of a related quiver, with the use of quiver Grassmannians.
The aim of the present paper is the computation of cluster variables in a quantum cluster algebra of finite type and of all cluster variables in an almost acyclic cluster. The result is achieved by considering the \(\mathbb F\)-valued representations of a quiver related to the quantum cluster algebra and by showing that an analog of the Caldero-Chapoton map holds when \(q\) is specialized to the order of \(\mathbb F\). Several examples illustrate the procedure.

16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
16T20 Ring-theoretic aspects of quantum groups
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