Cluster algebras and derived categories
Cluster algebras and derived categories
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DOI：
10.4171/115-1/6
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发表时间：
2012-02
期刊：
影响因子：
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通讯作者：
B. Keller
中科院分区：
文献类型：
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作者：
B. Keller
这是对簇代数及其使用金兹堡代数派生类别的（加法）分类的介绍性调查。在简要介绍簇组合学之后，我们回顾了接纳簇代数结构的坐标环的重要示例。然后，我们提出簇代数的一般定义，并描述簇变量、系数、c 向量和 g 向量之间的相互作用。我们展示了 c 向量如何出现在量子簇代数研究中以及它们与量子二对数的联系。然后，我们基于势颤的概念和相关金兹堡代数的派生范畴，提出了簇代数的加性分类框架。我们展示了先前介绍的组合学如何提升到分类水平，以及这如何导致与箭袋相关的簇代数证明福明-泽列文斯基的一些基本猜想。
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin-Zelevinsky's fundamental conjectures.