Introduction to Cluster Algebras
Introduction to Cluster Algebras
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DOI：
10.1007/978-3-319-56666-5_7
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
Max Glick;Dylan Rupel
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文献类型：
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作者：
Max Glick;Dylan Rupel
这些是2016年ASIDE会议上提出的一系列讲座的笔记。簇代数的定义是通过几个例子，即马尔可夫三元组，格拉斯曼，和双Bruhat细胞的出现在理论的总积极性的动机。一旦集群代数的定义是介绍了几个阶段的日益普遍性，证明基本结果勾勒在秩2的情况下。从这些基础上，我们建立了泊松结构的概念兼容的集群代数结构，并表明这将导致量化的集群代数。最后，我们给出了应用这些想法的形式Zamolodchikov周期性和五角星形映射的可积系统。
These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians, and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.