Indecomposable Representations of Graphs and Algebras
Indecomposable Representations of Graphs and Algebras
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DOI：
10.1090/memo/0173
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发表时间：
1976-06
期刊：
影响因子：
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通讯作者：
V. Dlab;C. Ringel
中科院分区：
文献类型：
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作者：
V. Dlab;C. Ringel
IN伯恩斯坦、IM Gelfand和VA Ponomarev最近证明了由P. Gabriel首先观察到的具有正定二次型的图的不可分解表示（“箭图”）与该形式的正根之间的双射可以直接证明。适当的函子从简单的表示产生所有不可分解的表示，就像外尔群的正则生成元从简单的表示产生所有正根一样。该方法在两个方向上扩展。为了处理所有的Dynkin图，而不是那些只有单边，我们认为有价值的图（“种”）。此外，我们考虑了半正定二次型的值图，即扩展Dynkin图。本文的主要结果描述了具有半正定二次型的值图的直到齐次表示为止的所有不可分解表示。这些不可分解的表示有两种类型：离散维类型的表示和连续维类型的表示。
IN Bernstein, IM Gelfand and VA Ponomarev have recently shown that the bijection, first observed by P. Gabriel, between the indecomposable representations of graphs (" quivers") with a positive definite quadratic form and the positive roots of this form can be proved directly. Appropriate functors produce all indecomposable representations from the simple ones in the same way as the canonical generators of the Weyl group produce all positive roots from the simple ones. This method is extended in two directions. In order to deal with all Dynkin diagrams rather than with those having single edges only, we consider valued graphs (" species"). In addition, we consider valued graphs with positive semi-definite quadratic form, ie extended Dynkin diagrams. The main result of the paper describes all indecomposable representations up to the homogeneous ones, of a valued graph with positive semi-definite quadratic form. These indecomposable representations are of two types: those of discrete dimension type, and those of continuous dimension type.