$2$-representation infinite algebras from non-abelian subgroups of ${\mathrm{SL}}_3$. Part I: Extensions of abelian groups

Abstract

Let $G\le {\mathrm{SL}}_3(\mathbb{C})$ 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 ${\mathrm{SL}}_3(\mathbb{C})$ 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 $G\simeq N⋊K$ 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.

Clarified the notion of cuts vs. gradings. Removed the incorrect Proposition 2.15 (v2) and adapted affected proofs. 28 pages. Comments welcome!