The Grothendieck Groups of Discrete Cluster Categories of Dynkin Type $A_{\infty }$

Abstract

In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty }$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.

arXiv commentV2: 18 pages, updated some figures, restated Theorem 4.4, comments welcome!