$T_{2}(T)$的几乎可裂序列及几乎$\mathcal {D}$-可裂序列
The Almost Split Sequences and ${\cal D}$-Split Sequences of $T_{2}(T)$

DOI：10.3770/j.issn:2095-2651.2013.01.002

 作者 单位 张玉林 北京工业大学应用数理学院, 北京 100124 姚海楼 北京工业大学应用数理学院, 北京 100124

AR箭图和导出等价是表示论的两个重要研究对象,AR序列与几乎$\mathcal{D}$-可裂序列是研究这两个对象的有力工具.因此研究特殊的下三角矩阵代数$T_{2}(T)=\begin{pmatrix} T & 0 \\ T & T \\ \end{pmatrix}$(其中$T$为域上的有限维代数)的表示,首先要确定它的AR序列与几乎$\mathcal{D}$-可裂序列.本文通过代数$T$的右(左)几乎可裂同态,既约同态,几乎可裂序列与几乎$\mathcal{D}$-可裂序列构造$T_{2}(T)$的相应同态及序列.并得到了一些有意义的结论.

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.