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- W2029223835 abstract "Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(begin{array}{cc}A&0 A&Aend{array})$ be the triangular matrix algebra and $A^{(1)}=(begin{array}{cc}A&0 DA&Aend{array})$ 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 ${rm rep.dim} T_2(A)$ is at most four if $A$ is not Dynkin type. Let $T$ be a tilting A-$module$ and $ol{T}=Toplusol{P}$ be a tilting $A^{(1)}$-$module$. We show that $End_{A^{(1)}} ol{T}$ is representation finite if and only if the full subcategory ${(X,Y,f) | Xin {rm mod} A, Yintau^{-1}mathscr{F}(T_A)cup{rm add} A}$ of ${rm mod T_2(A)}$ is of finite 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 at most three if $A$ is Dynkin type." @default.
- W2029223835 created "2016-06-24" @default.
- W2029223835 creator A5010335041 @default.
- W2029223835 creator A5010649438 @default.
- W2029223835 date "2011-07-19" @default.
- W2029223835 modified "2023-10-17" @default.
- W2029223835 title "Representation dimensions of triangular matrix algebras" @default.
- W2029223835 cites W1970058047 @default.
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- W2029223835 doi "https://doi.org/10.48550/arxiv.1107.3865" @default.
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