Representation Dimensions Of Triangular Matrix Algebras

Let A be a finite dimensional hereditary algebra over an algebraically closed field be the triangular matrix algebra and be the duplicated algebra of A respectively. We prove that rep.dim T2(A) is at most three if A is Dynkin type and rep.dim T2(A) is at most four if A is not Dynkin type. Let T be a tilting A-module and T=T(?)P be a tilting.A(1)-module. We show that EndA(1) T is representation finite if and only if the full subeategory{(X,Y,f)|X∈mod A,Y∈T-1F(TA) U add A} of mod T2(A) is of finite type, where r is the Auslander-Reiten translation and F(TA) is the torsion-free class of mod A associated with T. Moreover, we also prove that rep.dim End A(1) T is at most three if A is Dynkin type.Preprojective algebras is a kind of important algebra in representation theory, it is intensively related to Lie algebra and quantum group. In the last section of this paper, we give the mutation quiver of maximal rigid module of preprojective algebra A.A3.