Generalized McKay Quivers, Root System and Kac-Moody Algebras

Let $Q$ be a finite quiver and $G\subseteq\Aut(\mathbbm{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and $\Gamma$ is the generalized Mckay quiver and the valued graph corresponding to $(Q, G)$ respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $\mathfrak{g}(\Gamma)$. Moreover, we may lift $G$ to $\bar{G}\subseteq\Aut(\mathfrak{g}(\hat{Q}))$ such that $\mathfrak{g}(\Gamma)$ embeds into the fixed point algebra $\mathfrak{g}(\hat{Q})^{\bar{G}}$ and $\mathfrak{g}(\hat{Q})^{\bar{G}}$ as $\mathfrak{g}(\Gamma)$-module is integrable.