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

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 ${\mathcal{A}\mathcal{H}}_{|k|}(Q)$ 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.

12 pages