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

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... of SL(3, C) and of the finite subgroups obtained from embedding SL(2, C) into SL(3, C). 2020 Mathematics Subject Classification. Primary: 16G20, 16G60, 16S35; Secondary: 20C05. Proof. The first and the second assertions follow from Proposition 4.3 and Proposition 4.5 of [12], respectively. The last follows from the definition of Z Q. An n-properly-graded quiver is called n-nicely-graded quiver if it is nicely-graded. Proposition 2.2. Assume that Q is an n-nicely-graded quiver. Let Q be the returning arrow quiver of Q. Then Z| n−1 Q is isomorphic to a connected component of Z Q, hence is also nicely graded. Proof. Let d : Q 0 → Z be the function such that d(j) = d(i) + 1 for an arrow It is easy to see that φ is an isomorphism from Z| n−1 Q to the connected component of Z Q containing the vertex (i 0 , 0). By definition, an n-translation quiver is an (n + 1)-properly graded quiver. The bound quiver Z| n−1 Q is an (n + 1)-nicely graded quiver and the n-translation sends the ending vertex of a maximal bound path of length n + 1 to its starting vertex. So by the definition of complete τ -slice, we immediately have the following. Proposition 2.3. If Q is a stable n-translation quiver. Then any complete τ -slice in Z Q is an n-nicely-graded quiver. Let m ≤ l be two integers, write Z Q[m, l] for the full subquiver of Z Q with vertex set Z Q 0 [m, l] = {(i, t)|m ≤ t ≤ l}.