On A Class Of N-complete Algebras

Recently, Iyama introduced higher dimensional representation theory and studied n-representation finite and n-representation infinite. In fact,these higher dimensional representation theories are related to finite complexity self-injective algebras by Yoneda algebras and n-yame algebras. In this paper,we study a few of algebras related to higher dimensional representation theory. According to the tilting theory, we construct a class of n-complete algebras from the quiver of An with linear orientation; Higher Auslander algebras are a special class of n-complete algebras, we study the properties of these projec-tive, injective modules, and characterize the sufficient and necessary conditions such that projective modules are injective; At last, we prove that relationship of complexity and trivial extension.It mainly consists of the following two parts.N-complete algebras are a class of higher dimensional representation alge-bras with Tn-frinite. In Chapter 3, for linear orientation quiver of type An, by taking endomorphism rings of tilting module we construct a class of algebras ∧in,i=0,1, …,n--2, we proof these kinds of algebras are (i+1)-complete algebras. In particular, we also proof ∧nn-2 is an absolutely (n-1)-complete algebra, and ∧n+1n-1is the cone of ∧nn-2.First, we generalize the property of projective modules over the Yoneda algebra of Auslander algebra to higher dimensional cases in chapter 4. We discuss the necessary and sufficient conditions of a projective E(∧)op-module Px=(?)k=0n+1k∧e([X]∧/r) is injective over Yoneda algebra of higher dimensional Auslander algebra. Second,for a basic finite dimensional Koszul selfinjective al-gebra A over a field k, we show the relationship of the Yoneda algebra E(T(∧)) of trivial extension of A and the Yoneda algebra E(∧) of A. Complexity is a class of important invariants.Complexity and GK dimension are very impor-tance in studying of higher dimensional tame representation theory.At last,we proof that CT(∧1)= CT(∧2) if two finite dimensional k- algebras ∧1 and ∧2 are derived equivalence.