On quantum cluster algebras of finite type

TL;DR

This paper extends the quantum Caldero-Chapoton map to finite type quivers and proves that the algebra generated by all cluster characters coincides with the quantum cluster algebra.

Contribution

It introduces a quantum analogue of the Caldero-Chapoton map and establishes the equality of the generated algebra and the quantum cluster algebra for finite type quivers.

Findings

The algebra generated by all cluster characters equals the quantum cluster algebra for finite type quivers.

Extension of the Caldero-Chapoton map to quantum setting.

Proof of algebraic equality in finite type case.

Abstract

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined in \cite{rupel}. When $Q$ is a quiver of finite type, we prove that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$.