Cluster algebras
Cluster algebras
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DOI：
10.1073/pnas.1410635111
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发表时间：
2014-06
期刊：
影响因子：
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通讯作者：
B. Leclerc;L. Williams
中科院分区：
文献类型：
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作者：
B. Leclerc;L. Williams
簇代数是由Fomin和Zelevinsky（1）在2000年春天作为研究半单李群的对偶标准基和全正性的工具而提出的。然而，簇代数的理论从此开始了自己的生命，因为在数学的不同领域中已经发现了联系和应用，包括箭图和有限维代数的表示理论，参见，例如，参考文献。2、高血压病患者的心脏病发作次数为15次;泊松几何（16 - 19）;泰希穆勒理论（20 - 24）;弦理论（25 - 31）;离散动力系统和可积性（6，32 - 38）;组合学（39 - 47）。非常值得注意的是，集群代数提供了一个统一的代数和组合的框架，在这些和其他设置的各种现象。我们建议读者参考调查论文（36，48页）和集群代数门户网站（www.math.lsa.umich.edu/clustfomin/cluster.html），了解集群代数的各种介绍及其与数学（和物理）中其他学科的联系。简单地说，秩为k的簇代数A是k个变量的有理函数的环境域的子环，比如x1，...，xk。与大多数交换环不同，簇代数在一开始并不是通过一组完整的生成元和关系来表示的。相反，从初始种子的数据-包括k个初始聚类变量x 1，...，x k，加上一个交换矩阵-使用称为“突变”的迭代过程来产生其余的聚类变量。特别是，每一个新的集群.[1][2] 1应向其发送信函。电子邮件：威廉姆斯{at}math.berkeley.edu. [1]：#xref-corresp-1-1
Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1