On quantum cluster algebras of finite type

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 ${\mathcal{A}\mathcal{H}}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra ${\mathcal{EH}}_{|k|}(Q)$.

9 pages