Grothendieck groups of repetitive cluster categories

Abstract

In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories ${\mathcal{D}}^b(\mathcal{H})/⟨({\tau }^{-1}\Sigma {)}^p⟩$ for $1\le p\in \mathbb{N}$, where $\mathcal{H}$ is a hereditary abelian category with tilting objects. In this paper, we compute partial but essential results on the Grothendieck groups of the repetitive cluster categories ${\mathcal{D}}^b(\mathrm{mod}K{A}_n)/⟨({\tau }^{-1}\Sigma {)}^p⟩$ and ${\mathcal{D}}^b(\mathrm{mod}K{D}_n)/⟨({\tau }^{-1}\Sigma {)}^p⟩$. Our results extend the known computations for classical cluster categories, reveal new structural patterns arising from the repetitive parameter $p$, and provide further evidence of the close interplay between Grothendieck groups, Auslander-Reiten theory, and Coxeter transformations.

The previous version of this paper contained an error that affected the main results. In this revised version, we provide the corrected proofs of the results. In addition, a new coauthor has been added, who contributed substantially to identifying and resolving the issue