Representation dimensions of triangular matrix algebras

Abstract

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.dimT2(A) is at most three if A is Dynkin type and rep.dimT2(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 modT2(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.