通过理性泊松映射在GL_n上构造广义聚类结构
A generalized cluster structure on $GL_n$ via birational Poisson maps
AI总结：
本文通过理性泊松映射在GL_n上构造广义聚类结构，探讨了当非周期性条件不满足时的构造方法，结合了Cremmer-Gervais解与Weyl群最长元素的共轭关系。
摘要：
在最近的工作中，我们构造了一个从简单李群G到自身的有理映射，该映射将标准的泊松-李结构与由经典Yang-Baxter方程（CYBE）的 quasi-triangular 解（R-矩阵）定义的泊松齐次结构联系起来。在SL_n的情况下，如果与这些R-矩阵相关的组合Belavin-Drinfeld数据满足某些非周期性条件，该映射实际上是理性的，可以用来通过Berenstein-Fomin-Zelevinsky聚类变量的拉回获得SL_n上的 exotic 聚类结构。相同的策略后来被第一作者和D.Voloshyn用于描述与由quasi-triangular R-矩阵定义的泊松-李括号的泊松对偶兼容的广义聚类结构。在本文中，我们进一步推广使用理性的泊松映射来构造广义聚类结构，当非周期性条件不满足时应用该方法。为此，我们描述了在GL_n上与由两个Cremmer-Gervais解定义的泊松齐次括号兼容的广义聚类结构，这些解通过Weyl群最长元素的共轭相关。我们构造的关键成分是一个连接所考虑的括号与两个其他泊松括号的理性的映射：Cremmer-Gervais泊松-李括号在GL_{n-1}上的泊松对偶，以及一个与聚类代数解释的Coxeter-Toda流密切相关的复有理函数空间的括号。新的正则拉回种子和几乎聚类结构的概念也在我们的构造中发挥了重要作用。
英文摘要：
In a recent work, we constructed a rational map from a simple Lie group $\mathcal G$ to itself that intertwines the standard Poisson--Lie structure on $\mathcal G$ with a Poisson homogeneous one defined by a pair of quasi-triangular solutions to the classical Yang--Baxter equation (CYBE) known as R-matrices. We also showed, in the case of $SL_n$, that if the combinatorial Belavin--Drinfeld data associated with these R-matrices satisfies certain aperiodicity conditions, the map is, in fact, birational and can be used to obtain an initial cluster for an exotic cluster structure on $SL_n$ via the pullback of Berenstein--Fomin--Zelevinsky cluster variables. The same strategy was later used by the first author and D.~Voloshyn to describe generalized cluster structures compatible with the Poisson dual of the Poisson--Lie bracket defined by a quasi-triangular R-matrix. In this paper we further promote the use of birational Poisson maps in constructing generalized cluster structures by applying it in the situation when the aperiodicity condition is not satisfied. To this end, we describe a generalized cluster structure on $GL_n$ compatible with the Poisson homogeneous bracket defined by two Cremmer--Gervais solutions to the CYBE related via conjugation by the longest element of the Weyl group. The key ingredient to our construction is a birational map that connects the bracket under consideration with two other Poisson brackets: the Poisson dual to Cremmer--Gervais Poisson--Lie bracket on $GL_{n-1}$ and the bracket on a certain space of complex rational functions of one variable closely related to cluster algebraic interpretation of Coxeter--Toda flows. New notions of a regular pullback of a seed and of an almost-cluster structure whose detailed description are given in a separate note also play an important role in our construction.
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