On <inline-formula><tex-math id="M1">$ n $</tex-math></inline-formula>-slice algebras and related algebras

The \begin{document}$ n $\end{document} -slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of \begin{document}$ n $\end{document} -slice algebras via their \begin{document}$ (n+1) $\end{document} -preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame \begin{document}$ n $\end{document} -slice algebras to the McKay quiver of a finite subgroup of \begin{document}$ \mathrm{GL}(n+1, \mathbb C) $\end{document} . In the case of \begin{document}$ n = 2 $\end{document} , we describe the relations for the \begin{document}$ 2 $\end{document} -slice algebras related to the McKay quiver of finite Abelian subgroups of \begin{document}$ \mathrm{SL}(3, \mathbb C) $\end{document} and of the finite subgroups obtained from embedding \begin{document}$ \mathrm{SL}(2, \mathbb C) $\end{document} into \begin{document}$ \mathrm{SL}(3,\mathbb C) $\end{document} .