Grothendieck groups of repetitive cluster categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909813977382912 |
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| author | Chang, Huimin Murphy, Dave Zhou, Panyue |
| author_facet | Chang, Huimin Murphy, Dave Zhou, Panyue |
| contents | In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories $\mathcal D^b(\mathcal H)/\langle(τ^{-1}Σ)^p\rangle$ for $1\leq p\in\mathbb{N}$, 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 $\mathcal D^b({\rm mod}KA_n)/\langle(τ^{-1}Σ)^p\rangle$ and $\mathcal D^b({\rm mod} KD_n)/\langle(τ^{-1}Σ)^p\rangle$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grothendieck groups of repetitive cluster categories Chang, Huimin Murphy, Dave Zhou, Panyue Representation Theory Category Theory In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories $\mathcal D^b(\mathcal H)/\langle(τ^{-1}Σ)^p\rangle$ for $1\leq p\in\mathbb{N}$, 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 $\mathcal D^b({\rm mod}KA_n)/\langle(τ^{-1}Σ)^p\rangle$ and $\mathcal D^b({\rm mod} KD_n)/\langle(τ^{-1}Σ)^p\rangle$. 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. |
| title | Grothendieck groups of repetitive cluster categories |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2501.11021 |