Let $A$ be a finite dimensional hereditary algebra over an algebraicallyclosed field $k$, $T_2(A)=(\begin{array}{cc}A&0 A&A\end{array})$ be thetriangular matrix algebra and $A^{(1)}=(\begin{array}{cc}A&0 DA&A\end{array})$be the duplicated algebra of $A$ respectively. We prove that ${\rm rep.dim}\T_2(A)$ is at most three if $A$ is Dynkin type and ${\rm rep.dim}\ T_2(A)$ isat 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 fullsubcategory $\{(X,Y,f)\ |\ X\in {\rm mod}\ A,Y\in\tau^{-1}\mathscr{F}(T_A)\cup{\rm add}\ A\}$ of ${\rm mod \ T_2(A)}$ is offinite type, where $\tau$ is the Auslander-Reiten translation and$\mathscr{F}(T_A)$ is the torsion-free class of ${\rm mod}\ A$ associated with$T$. Moreover, we also prove that ${\rm rep.dim\ End}_{A^{(1)}}\ {\ol T}$ is atmost three if $A$ is Dynkin type.
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机译：假设$ A $是代数闭合域$ k $上的有限维遗传代数，$ T_2（A）=（\ begin {array} {cc} A＆0 A＆A \ end {array}）$是三角矩阵代数和$ A ^ {（1）} =（\ begin {array} {cc} A＆0 DA＆A \ end {array}）$分别是$ A $的重复代数。如果$ A $是Dynkin类型，我们证明$ {\ rm rep.dim} \ T_2（A）$最多为三个，如果$ A $，$ {\ rm rep.dim} \ T_2（A）$最多为四个。不是Dynkin类型。假设$ T $是倾斜的A-$ \ module $，而$ \ ol {T} = T \ oplus \ ol {P} $是倾斜的$ A ^ {（1）} $-$ \ module $。我们证明，当且仅当在{\ rm mod中的完整子类别$ \ {（X，Y，f）\ | \ X \时，$ \ End_ {A ^ {（1）}} \ ol {T} $是表示有限的} \ A，Y \ in \ tau ^ {-1} \ mathscr {F}（T_A）\ cup {\ rm add} \ A \} $ of $ {\ rm mod \ T_2（A）} $是有限类型，其中$ \ tau $是Auslander-Reiten翻译，而$ \ mathscr {F}（T_A）$是与$ T $关联的$ {\ rm mod} \ A $的无扭转类。此外，我们还证明，如果$ A $为Dynkin类型，则$ {\ rm rep.dim \ End} _ {A ^ {（1）}} \ {\ ol T} $最多为3。
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