Về đại số đường Leavitt của các đồ thị Hopf liên kết với nhóm hữu hạn

Abstract

This paper investigates the Hopf graphs associated with finite groups and their Leavitt path algebras. It shows that if G is a finite group and  $\mathfrak{r} = \sum_{C \in \mathcal{C}0} \mathfrak{r}_C C$  is a ramification data on G, then the semigroup S generated by the support of r is a normal subgroup of G. Based on this result, a decomposition result is established to describe the Leavitt path algebra $L_K(\Gamma_{G,\mathfrak{r}})$ as a direct sum of $\vert{}G\vert{} : \vert{}S\vert{}$ copies of  $L_K(\Gamma_{S,\mathfrak{r}})$. Furthermore, necessary and sufficient conditions are given on the ramification datas r for $L_K(\Gamma_{S,\mathfrak{r}})$ to be a purely infinite simple ring, and characterize when  $L_K(\Gamma_{G,\mathfrak{r}})$ is a simple, semi - simple ring in terms of the properties of the ramification datas $\mathfrak{r}$ and the subgroup S.