Cluster structures from 2-Calabi-Yau categories with loops

TL;DR

This paper extends the theory of cluster structures in 2-Calabi-Yau categories to include cases with loops, demonstrating their applicability to cluster categories of tubes and type B cluster algebras.

Contribution

It generalizes cluster structure axioms to include loops in endomorphism rings and applies this to model type B cluster algebra combinatorics.

Findings

Maximal rigid objects satisfy axioms with loops when no 2-cycles are present.

Cluster category of a tube models type B cluster algebra combinatorics.

Framework extends cluster theory to broader categorical settings.

Abstract

We generalise the notion of cluster structures from the work of Buan-Iyama-Reiten-Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi-Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.