The Grothendieck groups of n-cluster categories of type A_{\infty}

Abstract

In this article, we investigate the Grothendieck groups $K_0(\C_{A_{\infty}}^n)$ of $n$-cluster categories $\C_{A_{\infty}}^n$ of type $A_{\infty}$ introduced by T.~Holm and P.~J{\o}rgensen. We prove that $K_0(\C_{A_{\infty}}^n)\cong\mathbb{Z}$ for an arbitrary $n\geq 1$. As an application, this generalizes a result of Murphy for $n=1$.