In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n − 1)-translation algebras. We also recover Iyama's cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.

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Let Q be an acyclic quiver, it is classical that certain truncations of the translation quiver ZQ appear in the Auslander-Reiten quiver of the path algebra kQ. We introduce the n-translation quiver Z| n−1 Q as a generalization of the ZQ construction in our recent study on n-translation algebras. In this paper, we show that for certain algebra Γ of global dimension n, the quiver Z| n−1 Q can be used to describe the τ n -closure of DΓ and τ −1 n -closure of Γ in its module category and the νn-closure of Γ in the derived category.

Let Γ n be the cone of an (n − 1)-complete algebra over an algebraically closed field k. In this paper, we prove that if the bound quiver (Qn, ρn) of Γ n is a truncation from the bound McKay quiver (Q G , ρ G ) of a finite subgroup G of GL(n, k), the bound quiver (Q n+1 , ρ n+1 ) of Γ n+1 , the cone of Γ n , is a truncation from the bound McKay quiver (Recently, Iyama introduced n-cluster tilting subcategories and developed higher Auslander-Reiten theory ([12]). In [10], he introduced and characterized a class of higher representation algebras, n-complete algebras. Such algebras are preserved under cone constructions. He also proved that n-Auslander absolutely n-complete algebra are constructed by iterative cone construction starting from some path algebra of quiver of type A r with linear orientation. Guo proved that such algebras can be obtained from a truncation of the McKay quivers of some abelian groups ([5]).In this paper, we generalize the results of Guo and prove the following result: Let Γ n be the cone of an (n − 1)-complete algebra, if the bound quiver (Q n , ρ n ) of Γ n is a truncation from the bound McKay quiver (Q G , ρ G ) of a finite subgroup G of GL(n, k), then there exists a positve integer m such that the bound quiver (Q n+1 , ρ n+1 ) of Γ n+1 is a truncation from the bound McKay quiver (Q G , ρ G ) of a finite subgroup G ∼ = G × Z m in GL(n + 1, k).The paper is organized as follows. In Section 1, we shall recall some basic definitions and facts needed for n-complete algebras, McKay quivers and trivial extensions of graded self-injective algebras. Then we describe the bound McKay quivers using twisted trivial extensions in Section 2 and Section 3. Our main theorem will be stated and proved in Section 4. PreliminariesThroughout this paper, k is an algebraically closed field of characteristic 0. Let Λ be an algebra over k. Denote by mod Λ the category of finitely generated left Λ-modules, and for M ∈ mod Λ, denote by add M the full subcategory of mod Λ consisting of direct summands of finite direct sums of copies of M , and denote by D the standard k-duality Hom k (−, k).

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