Plethora of cluster structures on $GL_n$

Plethora of cluster structures on $GL_n$
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$GL_n$ 上有过多的簇结构

DOI：
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发表时间：
2019
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通讯作者：
A. Vainshtein
A. Vainshtein
中科院分区：
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文献类型：
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作者：
M. Gekhtman;M. Shapiro;A. Vainshtein

文献摘要

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我们继续研究了$GL_n$, $SL_n$和$operatorname{Mat}_n$上正则函数环上兼容泊松-李和泊松-齐次结构的多簇结构。根据我们最初的猜想，半单复群$mathcal G$上泊松—李结构的Belavin-Drinfeld分类中的每一类对应于$mathcal O(mathcal G)$中的一个簇结构。在此，我们用一个包含所有已知示例的$A_n$类型的Belavin-Drinfeld （BD）数据的大子集证明了这个猜想。也就是说，我们将所有可能的$A_n$类型BD数据细分为面向和非面向类型。在有向情况下，我们挑选出满足非周期性组合条件的BD数据，并证明对于任何这类BD数据都存在与相应泊松-李括号兼容的规则簇结构。事实上，我们将非周期条件推广到有取向的BD数据对上，并证明了一个更一般的结果，即在$SL_n$上存在一个规则的簇结构，该簇结构与泊松括号齐次兼容，对于两个具有不同泊松-李括号的$SL_n$副本的左右作用是齐次的。如果不满足非周期性条件，则必须将兼容的簇结构替换为广义的簇结构。我们将在以后的出版物中讨论这种情况。
We continue the study of multiple cluster structures in the rings of regular functions on $GL_n$, $SL_n$ and $operatorname{Mat}_n$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $mathcal G$ corresponds to a cluster structure in $mathcal O(mathcal G)$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $A_n$ type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $SL_n$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $SL_n$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications.