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

TL;DR

This paper calculates the Grothendieck groups for discrete cluster categories of Dynkin type A_infinity and their completions, providing algebraic invariants for these categories.

Contribution

It explicitly determines the Grothendieck groups for the discrete cluster categories of type A_infinity and their completions, extending previous work in the area.

Findings

Computed the triangulated Grothendieck groups for discrete cluster categories of type A_infinity.

Determined the Grothendieck groups of the completions of these categories.

Extended the understanding of algebraic invariants in cluster categories.

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.