Quotients of cluster categories

TL;DR

This paper demonstrates that in Dynkin types A and D, higher cluster categories are essentially quotients of cluster or 2-cluster categories, revealing that their phenomena are embedded within these simpler categories, unlike in type E.

Contribution

It shows that in Dynkin types A and D, higher cluster categories can be obtained as quotients of simpler categories, simplifying their understanding.

Findings

Half of higher cluster categories in types A and D are quotients of cluster categories.

The other half are quotients of 2-cluster categories.

In type E, higher cluster phenomena are not reducible to simpler categories.

Abstract

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other half can be obtained as quotients of 2-cluster categories, the "lowest" type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.