Representation dimensions of triangular matrix algebras

Abstract

Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(\begin{array}{ccc}A& 0A& A\end{array})$ be the triangular matrix algebra and $A^{(1)}=(\begin{array}{ccc}A& 0DA& A\end{array})$ be the duplicated algebra of $A$ respectively. 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 $T$ be a tilting A-$\module$ and $\ol{T}=T\oplus\ol{P}$ be a tilting $A^{(1)}$-$\module$. We show that $\End_{A^{(1)}} \ol{T}$ is representation finite if and only if the full subcategory $\{(X,Y,f)|X\in \mathrm{mod}A,Y\in {\tau }^{-1}\mathcal{F}(T_A)\cup \mathrm{add}A\}$ of $\mathrm{mod}{\mathrm{T}}_2(\mathrm{A})$ is of finite type, where $\tau$ is the Auslander-Reiten translation and $\mathcal{F}(T_A)$ is the torsion-free class of $\mathrm{mod}A$ associated with $T$. Moreover, we also prove that ${\rm rep.dim\ End}_{A^{(1)}}\ {\ol T}$ is at most three if $A$ is Dynkin type.

19 pages