American Institute of Mathematical Sciences

Let \begin{document}$ \mathcal{G}(\mathcal{X}) $\end{document} and \begin{document}$ \mathcal{G}(\mathcal{Y}) $\end{document} be Gorenstein subcategories induced by an admissible balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document} in an abelian category \begin{document}$ \mathcal{A} $\end{document}. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of \begin{document}$ \mathcal{A} $\end{document} induced by the balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document}. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring \begin{document}$ R $\end{document}.

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}.