$n$-complete algebras and McKay quivers

TL;DR

This paper explores the relationship between cones of $(n-1)$-complete algebras and McKay quivers, showing how the quivers of higher cones relate to McKay quivers of extended groups.

Contribution

It establishes that the bound quiver of the cone of an $(n-1)$-complete algebra can be obtained as a truncation from a McKay quiver of an extended group, generalizing the connection.

Findings

The bound quiver of the cone of an algebra is a truncation of a McKay quiver.

Extension of the group by a cyclic group corresponds to the cone operation on the algebra.

The relationship holds inductively for successive cones of the algebra.

Abstract

Let $\Gamma^{n}$ be the cone of an $(n-1)$-complete algebra over an algebraically closed field $k$. In this paper, we prove that if the bound quiver $(Q_{n},\rho_{n})$ of $\Gamma^{n}$ is a truncation from the bound McKay quiver $(Q_{G},\rho_{G})$ of a finite subgroup $G$ of $GL(n,k)$, the bound quiver $(Q_{n+1}, \rho_{n+1})$ of $\Gamma^{n+1}$, the cone of $\Gamma^{n}$, is a truncation from the bound McKay quiver $(Q_{\widetilde{G}},\rho_{\widetilde{G}})$ of $\widetilde{G}$, where $\widetilde{G}\cong G\times \mathbb{Z}_{m}$ for some $m\in \mathbb{N}$.