On $n$-slice Algebras and Related Algebras

TL;DR

This paper introduces and classifies $n$-slice algebras, connecting them to higher preprojective algebras, trivial extensions, and McKay quivers, with specific results for the case when n=2.

Contribution

It provides a classification framework for $n$-slice algebras using their $(n+1)$-preprojective algebras and quadratic duals, extending higher dimensional representation theory.

Findings

Classification of $n$-slice algebras via preprojective algebras and trivial extensions.

Relation of tame $n$-slice algebras to McKay quivers of finite subgroups.

Explicit description of relations for 2-slice algebras related to specific finite subgroups.

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