Let A be a finite dimensional hereditary algebra over an algebraically closed field k,T2(A)=A0AA be the triangular matrix algebra. 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 A(~1)=A0DAA be the duplicated algebra of A. 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 subcategory {(~(XA,YA),f)|~(XA)∈modA, ~(YA)∈τ-~1F(~(TA))∪addA} of mod ~(T2)(A) is of finite type, where τ is the Auslander-Reiten translation and F(~(TA)) is the torsion-free class of modA associated with T. Moreover, we also prove that rep.dimEndA(_1)T is at most three if A is Dynkin type.
展开▼
