The Almost Split Sequences and ${\cal D}$-Split Sequences of $T_{2}(T)$
Received:July 15, 2011  Revised:October 31, 2011
Key Words: algebras   modules   triangular matrix algebras   AR sequences   approximations.
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10971172) and the Natural Science Foundation of Beijing (Grant Nos.1092002; 1122002).
 Author Name Affiliation Yulin ZHANG College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China Hailou YAO College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China
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The AR-quiver and derived equivalence are two important subjects in the representation theory of finite dimensional algebras, and for them there are two important research tools-AR-sequences and ${\cal D}$-split sequences. So in order to study the representations of triangular matrix algebra $T_{2}(T)=\begin{pmatrix} T & 0 \\ T & T \\ \end{pmatrix}$ where $T$ is a finite dimensional algebra over a field, it is important to determine its AR-sequences and ${\cal D}$-split sequences. The aim of this paper is to construct the right(left) almost split morphisms, irreducible morphisms, almost split sequences and ${\cal D}$-split sequences of $T_{2}(T)$ through the corresponding morphisms and sequences of $T$. Some interesting results are obtained.