Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces
Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces
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DOI：
10.1016/j.aim.2017.03.016
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发表时间：
2014-08
期刊：
影响因子：
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通讯作者：
Ilke Çanakçi;Sibylle Schroll
中科院分区：
文献类型：
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作者：
Ilke Çanakçi;Sibylle Schroll
在有限维代数表示理论的背景下，弦代数得到了广泛的研究，其表示理论的大多数方面都得到了很好的理解。一个例外是不可分解模块之间的扩展分类。在本文中，我们明确地描述了一类弦代数的这种扩展，即与曲面三角化相关的温和代数。这些代数产生为未穿孔曲面的雅可比代数。我们将不可分解模的扩展空间与曲面上的广义弧相交联系起来，并给出了几乎所有情况下不可分解模的扩展空间的显式基。我们证明了这些扩展空间的尺寸是用曲面上的交叉弧来表示的。我们的方法是新的，包括将蛇形图解释为不可分解的模块。为了证明我们的基是一个生成集，我们需要在相关的聚类范畴中工作，其中我们显式地计算扩展的中间项并给出其扩展空间的基。我们注意到并不是所有在聚类范畴中的扩展都会引起雅可比代数的扩展。
In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of generalised arcs in the surface and give explicit bases of the extension spaces for indecomposable modules in almost all cases. We show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface.Our approach is new and consists of interpreting snake graphs as indecomposable modules. In order to show that our basis is a spanning set, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give bases of their extension spaces. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.