n-complete algebras and McKay quivers

Abstract

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).

Higher Zigzag Algebras

“…here. See also the related papers [ZLZ17] and [Guo16], which also study twisted trivial extensions, skew group algebras, and McKay quivers.…”