张玉林,姚海楼.$T_{2}(T)$的几乎可裂序列及几乎$\mathcal {D}$-可裂序列[J].数学研究及应用,2013,33(1):11~22
$T_{2}(T)$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 $T_{2}(T)$
投稿时间:2011-07-15  最后修改时间:2011-10-31
DOI:10.3770/j.issn:2095-2651.2013.01.002
中文关键词:  代数    下三角矩阵代数  AR序列  逼近.
英文关键词:algebras  modules  triangular matrix algebras  AR sequences  approximations.
基金项目:国家自然科学基金(Grant No.10971172); 北京市自然科学基金(Grant Nos.1092002; 1122002).
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 
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中文摘要:
      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.
英文摘要:
      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.
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