Cluster structures from 2-Calabi-Yau categories with loops

Aslak Bakke Buan, Bethany Marsh, Dagfinn F. Vatne
Arxiv ID: 0810.3132Last updated: 12/21/2020
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.

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