On $n$-slice Algebras and Related Algebras

Abstract

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 $\mathrm{SL}(3,\mathbb{C})$.

References in corpus (1)

• On McKay Quiver and Covering Spaces