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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.24707 |
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| _version_ | 1866917409118486528 |
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| author | Söderberg, Christoffer |
| author_facet | Söderberg, Christoffer |
| contents | Buan, Iyama, Reiten and Smith proved that cluster-tilting objects in triangulated 2-Calabi--Yau categories are closely connected with mutation of quivers with potentials over an algebraically closed field. We prove a more general statement where instead of working with quivers with potentials we consider species with potential over a perfect field.
We describe the $3$-preprojective algebra of the tensor product of two tensor algebras of acyclic species using a species with potential. In the case when the Jacobian algebra of a species with potential is self-injective, we provide a description of the Nakayama automorphism of a particular case of mutation of the species with potential where you mutate along orbits of the Nakayama permutation, which preserves self-injectivity.
For certain types of Jacobian algebras of species with potentials, we prove that they lie in the scope of the derived Auslander-Iyama correspondence due to Jasso-Muro. Mutating along orbits of the Nakayama permutation stays within this setting, yielding a rich source of examples. All $2$-representation finite $l$-homogeneous algebras that are constructed using certain species with potential and mutations of such species with potentials are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mutating Species with Potentials and Cluster Tilting Objects Söderberg, Christoffer Representation Theory Buan, Iyama, Reiten and Smith proved that cluster-tilting objects in triangulated 2-Calabi--Yau categories are closely connected with mutation of quivers with potentials over an algebraically closed field. We prove a more general statement where instead of working with quivers with potentials we consider species with potential over a perfect field. We describe the $3$-preprojective algebra of the tensor product of two tensor algebras of acyclic species using a species with potential. In the case when the Jacobian algebra of a species with potential is self-injective, we provide a description of the Nakayama automorphism of a particular case of mutation of the species with potential where you mutate along orbits of the Nakayama permutation, which preserves self-injectivity. For certain types of Jacobian algebras of species with potentials, we prove that they lie in the scope of the derived Auslander-Iyama correspondence due to Jasso-Muro. Mutating along orbits of the Nakayama permutation stays within this setting, yielding a rich source of examples. All $2$-representation finite $l$-homogeneous algebras that are constructed using certain species with potential and mutations of such species with potentials are considered. |
| title | Mutating Species with Potentials and Cluster Tilting Objects |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2509.24707 |