$n$-complete algebras and McKay quivers

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 $(Q_{\tilde{G}},{\rho }_{\tilde{G}})$ of $\tilde{G}$, where $\tilde{G}\cong G\times {\mathbb{Z}}_m$ for some $m\in \mathbb{N}$.

13 pages