On Quantum Analogue of The Caldero-Chapoton Formula

Abstract

Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We show that an analogue of the Caldero-Chapoton formula holds for all quantum cluster algebras of finite type and for any cluster variable in an almost acyclic cluster.