On $n$-slice Algebras and Related Algebras

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