Integrable systems on quad-graphs
Integrable systems on quad-graphs
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DOI：
10.1155/s1073792802110075
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发表时间：
2001-10
影响因子：
1
通讯作者：
A. Bobenko;Y. Suris
中科院分区：
文献类型：
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作者：
A. Bobenko;Y. Suris
离散（格）系统构成了可积系统理论的一个完善的部分。他们已经出现在早期的理论（见，例如[11，12]），并逐步越来越重要的地方，它（比照。[18]中的评论）。如今，许多专家在该领域同意，离散可积系统在许多方面甚至比连续系统更基本。它们在可积系统的各种应用中起着突出的作用，例如离散微分几何（参见，例如，[9]中的评论）。传统上，离散可积系统的独立变量被认为属于一个正则正方形格Z（或其多维类似物Z）。直到最近，才出现了第一个线索，存在一个丰富的和有意义的理论，可积系统的非正方形格，更一般地说，对任意图形。相关的出版物几乎被[2，3，5，6，16，20，21，22]穷尽。我们将图上的可积系统定义为具有圈群中的值的平坦连接。这是非常自然的定义，离散可积系统的专家不仅会立即接受它，甚至可能认为它微不足道。然而，它只是最近才结晶，似乎没有出现在文献[3，5，6]之前。(It应该注意的是，Novikov与合作者正在开发图上可积系统的不同框架[16，20，21]。我们（与Hoffmann一起）将圆模式作为离散复分析的对象进行研究，从而导致我们考虑这样的系统：在[5，6]中，我们证明了具有正六边形晶格组合学的某些类圆模式
Discrete (lattice) systems constitute a well-established part of the theory of integrable systems. They came up already in the early days of the theory (see, e.g. [11, 12]), and took gradually more and more important place in it (cf. a review in [18]). Nowadays many experts in the field agree that discrete integrable systems are in many respects even more fundamental than the continuous ones. They play a prominent role in various applications of integrable systems such as discrete differential geometry (see, e.g., a review in [9]). Traditionally, independent variables of discrete integrable systems are considered as belonging to a regular square lattice Z (or its multidimensional analogs Z). Only very recently, there appeared first hints on the existence of a rich and meaningful theory of integrable systems on nonsquare lattices and, more generally, on arbitrary graphs. The relevant publications are almost exhausted by [2, 3, 5, 6, 16, 20, 21, 22]. We define integrable systems on graphs as flat connections with the values in loop groups. This is very natural definition, and experts in discrete integrable systems will not only immediately accept it, but might even consider it trivial. Nevertheless, it crystallized only very recently, and seems not to appear in the literature before [3, 5, 6]. (It should be noted that a different framework for integrable systems on graphs is being developed by Novikov with collaborators [16, 20, 21].) We were led to considering such systems by our (with Hoffmann) investigations of circle patterns as objects of discrete complex analysis: in [5, 6] we demonstrated that certain classes of circle patterns with the combinatorics of regular hexagonal lattice