On Quantum Analogue of The Caldero-Chapoton Formula

TL;DR

This paper extends the Caldero-Chapoton formula to quantum cluster algebras of finite type, establishing a link between valued quiver representations and cluster variable expansions in the quantum setting.

Contribution

It introduces a quantum analogue of the Caldero-Chapoton formula applicable to all finite type quantum cluster algebras and almost acyclic clusters.

Findings

The quantum Caldero-Chapoton formula holds for finite type quantum cluster algebras.

Connections are established between valued quiver representations and quantum cluster variable expansions.

The results generalize classical formulas to the quantum algebra context.

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.