A classification of $n$-representation infinite algebras of type Ã
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913583089057792 |
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| author | Dramburg, Darius Gasanova, Oleksandra |
| author_facet | Dramburg, Darius Gasanova, Oleksandra |
| contents | We classify $n$-representation infinite algebras $Λ$ of type Ã. This type is defined by requiring that $Λ$ has higher preprojective algebra $Π_{n+1}(Λ) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq \operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a Dimer model. In terms of toric geometry and McKay correspondence, the types form a lattice simplex of junior elements of $G$. We show that all algebras of the same type are related by iterated $n$-APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_06553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A classification of $n$-representation infinite algebras of type à Dramburg, Darius Gasanova, Oleksandra Representation Theory 16G20, 16S35, 16E65, 14E16 We classify $n$-representation infinite algebras $Λ$ of type Ã. This type is defined by requiring that $Λ$ has higher preprojective algebra $Π_{n+1}(Λ) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq \operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a Dimer model. In terms of toric geometry and McKay correspondence, the types form a lattice simplex of junior elements of $G$. We show that all algebras of the same type are related by iterated $n$-APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements. |
| title | A classification of $n$-representation infinite algebras of type à |
| topic | Representation Theory 16G20, 16S35, 16E65, 14E16 |
| url | https://arxiv.org/abs/2409.06553 |