Cluster Morita theorem for negative cluster categories
Abstract
Fix an integer d≤−2. We characterize Hom-finite algebraic triangulated categories admitting a (−d)-simple-minded system as stable categories underlineCM(B) of proper (−d)-self-injective non-positive dg algebras; equivalently, each admits a d-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a d-Calabi--Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right (d+1)-Calabi--Yau structures on the finite-dimensional part with normalized right d-Calabi--Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav--Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left--right Calabi--Yau symmetry.
Related
- Auslander correspondence for higher stable dg categories and cluster Morita theory
- Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
- Negative correlation for the random-cluster model below one on high-degree regular graphs
- Negative Correlations for Forests and the q<1 Random Cluster Model
- Exact dg categories II: The embedding theorem
BibTeX
@misc{fushimi2026cluster,
title = {{Cluster Morita theorem for negative cluster categories}},
author = {Riku Fushimi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10396},
eprint = {2609.10396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10396},
}
RIS
TY - UNPB
TI - Cluster Morita theorem for negative cluster categories
AU - Riku Fushimi
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10396
UR - https://arxiv.org/abs/2609.10396
AB - Fix an integer d≤−2. We characterize Hom-finite algebraic triangulated categories admitting a (−d)-simple-minded system as stable categories underlineCM(B) of proper (−d)-self-injective non-positive dg algebras; equivalently, each admits a d-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a d-Calabi--Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right (d+1)-Calabi--Yau structures on the finite-dimensional part with normalized right d-Calabi--Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav--Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left--right Calabi--Yau symmetry.
ER -