Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors
Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors
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DOI：
10.1007/bfb0088461
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发表时间：
1980
期刊：
影响因子：
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通讯作者：
S. Brenner;M. C. R. Butler
中科院分区：
文献类型：
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作者：
S. Brenner;M. C. R. Butler
引言 Bernstein、Gelfand 和 Ponomarev 在他们关于 4 子空间问题和加布里埃尔定理的工作中将反射函子引入到箭袋表示论中，并且有一些推广，请参见 [13]、[6]、[10] 和 [2]。本文的目的是提出该概念的进一步扩展，并为带有关系的箭袋（QWR）提供一些应用。 Marmaridis [19] 开发了该理论的一个特例，并将其应用于某些 QWR；事实上，他的论文[18]中使用的一些方法也可以被视为这些函子的应用，尽管它们没有以这种方式呈现。与箭袋的任何表示相关联的是维度向量，不可分解模块的维度向量是与箭袋相关的二次形式的正根（参见例如[6]，[10]，[15]）。类似的结果似乎也适用于某些 QWR。反射函子的一些应用涉及对其引起的维度向量的变换的研究。事实证明，我们的函子有一些应用，它们利用了类似的变换，我们喜欢将其视为固定根系统基础的变化 - 轴相对于根的倾斜，导致位于正锥体中的根的不同子集。（在第 4 章第 2 节中详细考虑了一个例子）。出于这个原因，并且因为“倾斜”这个词很容易变形，我们将我们的函子称为倾斜函子或简称为倾斜。
I nt r oduc ti on Reflection functors were introduced into the representation theory of quivers by Bernstein, Gelfand and Ponomarev in their work on the 4-subspace problem and on Gabriel's Theorem and there have been several generalisations, see [13],[6],[10] and [2]. The aim of this paper is to present a further extension of the concept and to give some applications to quivers with relations (QWR's). A special case of this theory has been developed by Marmaridis [19] and applied to certain QWR's; indeed some of the methods used in his Thesis [18] may also be regarded as applications of these functors, though they are not presented in that way.Associated with any representation of a quiver is a dimension vector, and the dimension vectors of indecomposable modules are the positive roots of the quadratic form associated to the quiver (see eg [6],[10],[15]). Similar results seem to hold for certain QWR's. Some applications of reflection functors involve the study of the transformations of dimension vectors they induce. It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed rootsystem-a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone.(An example is considered in some detail in Chapter 4, § 2). For this reason, and because the word'tilt'inflects easily, we call our functors tilting functors or simply tilts.