Journal of the
Korean Mathematical Society JKMS

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

Keywords: generalized McKay quiver, representation of quiver, root system, Kac-Moody algebra

MSC numbers: Primary 16G10, 16G20, 17B67