Representation dimensions of triangular matrix algebras

Abstract Let A be a finite dimensional hereditary algebra over an algebraically closed field k , T 2 ( A ) = A 0 A A be the triangular matrix algebra. We prove that rep . dim T 2 ( A ) is at most three if A is Dynkin type and rep . dim T 2 ( A ) is at most four if A is not Dynkin type. Let A ( 1 ) = A 0 DA A 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 End A ( 1 ) T ¯ is representation finite if and only if the full subcategory { ( X A , Y A , f ) | X A ∈ mod A , Y A ∈ τ - 1 F ( T A ) ∪ add A } of mod T 2 ( A ) is of finite type, where τ is the Auslander–Reiten translation and F ( T A ) 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.