On a quantum analog of the Caldero-Chapoton formula. (English) Zbl 1237.16013

Rupel, Dylan

Int. Math. Res. Not. 2011, No. 14, 3207-3236 (2011).

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.

Reviewer: Giovanna Carnovale (Padova)

MSC:

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

Keywords:

cluster variables; quantum cluster algebras; representations of valued quivers; quiver Grassmannians