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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.19582 |
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| _version_ | 1866914495412043776 |
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| author | Chan, Aaron Iyama, Osamu Marczinzik, Rene |
| author_facet | Chan, Aaron Iyama, Osamu Marczinzik, Rene |
| contents | We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractionally Calabi-Yau algebras and cluster tilting Chan, Aaron Iyama, Osamu Marczinzik, Rene Representation Theory We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective algebra. |
| title | Fractionally Calabi-Yau algebras and cluster tilting |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2604.19582 |