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
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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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
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