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 GL(n + 1, 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 SL(3, C) and of the finite subgroups obtained from embedding SL(2, C) into SL(3, C).