Multiplicative properties of a quantum Caldero-Chapoton map associated to valued quivers

TL;DR

This paper proves a multiplication theorem for a quantum Caldero-Chapoton map related to valued quivers, extending previous results and establishing the algebra generated by cluster characters as the quantum cluster algebra in specific cases.

Contribution

It introduces a multiplication theorem for the quantum Caldero-Chapoton map associated to valued quivers, expanding the understanding of quantum cluster algebras.

Findings

The algebra generated by all cluster characters equals the quantum cluster algebra for finite type or rank 2 valued quivers.

Various bases of quantum cluster algebras of rank 2 are derived.

The multiplication theorem extends previous results in the literature.

Abstract

We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in \cite{DX}\cite{D}. As an application, when $Q$ is a valued quiver of finite type or rank 2, we obtain that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$ and various bases of the quantum cluster algebras of rank 2 can naturally be deduced.