### Electronic Research Archive

2021, Issue 4: 2687-2718. doi: 10.3934/era.2021009
Special Issues

# On $n$-slice algebras and related algebras

• Received: 01 April 2020 Revised: 01 December 2020 Published: 01 September 2021
• Primary: 16G20, 16G60, 16S35; Secondary: 20C05

• 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 $\mathrm{SL}(3,\mathbb C)$.

Citation: Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $n$-slice algebras and related algebras[J]. Electronic Research Archive, 2021, 29(4): 2687-2718. doi: 10.3934/era.2021009

### Related Papers:

• 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 $\mathrm{SL}(3,\mathbb C)$.

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