N-Complete Algebras And McKay Quiver

Self-injective algebras play an important role in the representation theory of algebras,actually,important and interesting examples of self-injective algebras are provided by trivial extensions of finite dimensional algebras and twisted trivial extensions of finite dimensional algebras.In this paper,as a generalization of the Morita equivalence of trivial extensions of finite dimensional algebras,we prove that the Morita equivalence of twisted trivial extensions of finite dimensional algebras.Furthermore,we describe the isomorphism between twisted trivial extension algebras and twisted tensor product algebras,as an application of the isomorphism,we give the bound quiver of twisted trivial extensions of the self-injective algebras.Recently,Iyama introduced higher representation theory,such as:n-Auslander algebras,n-Auslander-Reiten translation functor,Iyama introduced n-cluster tilting subcategories and developed higher Auslander-Reiten theory.He introduced and characterized a class of higher representation algebras,ncomplete algebras.Such algebras are preserved under cone constructions.He also proved that n-Auslander absolutely n-complete algebras are constructed by iterative cone construction starting from some path algebra of quiver of type Ar with linear orientation.Guo proved that such algebras can be obtained from an truncation of the McKay quivers of some abelian groups.We generalize the results of Guo and prove that the following result:Let?ⁿ be the cone of an?n-1?-complete algebra,if the bound quiver?Q_n,?_n?of?ⁿ is a truncation from the bound McKay quiver?Q_G,?_G?of a finite subgroup G of GL?n,k?,then there exists an positve integer m such that the bound quiver(Q_n+1,?_n+1)of ?ⁿ⁺¹ is a truncation from the bound McKay quiver?Q_G,?_G?of a finite subgroup ????G × Z_m in GL?n+1,k?.