{
  "id": "2505.10683",
  "version": "v1",
  "published": "2025-05-15T19:59:22.000Z",
  "updated": "2025-05-15T19:59:22.000Z",
  "title": "$2$-representation infinite algebras from non-abelian subgroups of $\\operatorname{SL}_3$. Part I: Extensions of abelian groups",
  "authors": [
    "Darius Dramburg",
    "Oleksandra Gasanova"
  ],
  "comment": "19 pages. Comments welcome!",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "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 structure. We show that the algebra $R \\ast G$ admits a $3$-preprojective structure 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-15T19:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16S35"
    ],
    "keywords": [
      "representation infinite algebras",
      "abelian group",
      "non-abelian subgroups",
      "preprojective structure",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}