$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups
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| Format: | Preprint |
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2025
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| _version_ | 1866908851415023616 |
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| author | Dramburg, Darius Gasanova, Oleksandra |
| author_facet | Dramburg, Darius Gasanova, Oleksandra |
| contents | Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. The resulting skew-group algebra $R \ast G$ is $3$-Calabi-Yau, and can sometimes be endowed with the structure of a $3$-preprojective algebra. However, not every such $R \ast G$ admits such a structure. The finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ are classified into types (A) to (L). We consider the groups $G$ of types (C) and (D) and determine for each such group whether the algebra $R \ast G$ admits a $3$-preprojective cut, that is a $3$-preprojective structure arising from a grading of the McKay quiver of $G$. We show that the algebra $R \ast G$ admits a $3$-preprojective cut if and only if $9 \mid |G|$. Our proof is constructive and yields a description of the involved $2$-representation infinite algebras. This is based on the semi-direct decomposition $G \simeq N \rtimes K$ for an abelian group $N$, and we show that the existence of a $3$-preprojective structure on $R \ast G$ is essentially determined by the existence of one on $R \ast N$. This provides new classes of $2$-representation infinite algebras, and we discuss some $2$-Auslander-Platzeck-Reiten tilts. Along the way, we give a detailed description of the involved groups and their McKay quivers by iteratively applying skew-group constructions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_10683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups Dramburg, Darius Gasanova, Oleksandra Representation Theory Rings and Algebras 16G20, 16S35 Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. The resulting skew-group algebra $R \ast G$ is $3$-Calabi-Yau, and can sometimes be endowed with the structure of a $3$-preprojective algebra. However, not every such $R \ast G$ admits such a structure. The finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ are classified into types (A) to (L). We consider the groups $G$ of types (C) and (D) and determine for each such group whether the algebra $R \ast G$ admits a $3$-preprojective cut, that is a $3$-preprojective structure arising from a grading of the McKay quiver of $G$. We show that the algebra $R \ast G$ admits a $3$-preprojective cut if and only if $9 \mid |G|$. Our proof is constructive and yields a description of the involved $2$-representation infinite algebras. This is based on the semi-direct decomposition $G \simeq N \rtimes K$ for an abelian group $N$, and we show that the existence of a $3$-preprojective structure on $R \ast G$ is essentially determined by the existence of one on $R \ast N$. This provides new classes of $2$-representation infinite algebras, and we discuss some $2$-Auslander-Platzeck-Reiten tilts. Along the way, we give a detailed description of the involved groups and their McKay quivers by iteratively applying skew-group constructions. |
| title | $2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups |
| topic | Representation Theory Rings and Algebras 16G20, 16S35 |
| url | https://arxiv.org/abs/2505.10683 |