Log-canonical coordinates for symplectic groupoid and cluster algebras
Log-canonical coordinates for symplectic groupoid and cluster algebras
复制标题
辛群群和簇代数的对数正则坐标
DOI：
10.1093/imrn/rnac101
复制
发表时间：
2022
影响因子：
1
通讯作者：
Shapiro, M.
中科院分区：
文献类型：
--
作者：
Chekhov, L.;Shapiro, M.
利用Fock-Goncharov高阶Teichmüler空间变量，我们导出了上三角矩阵的群组的一般辛叶的对数正则坐标表示，以及在更一般的情况下，由三角矩阵的反射方程控制的代数的高维辛叶的对数正则坐标表示.所得结果与以前得到的带孔Riemann曲面的群变量的Poisson和量子表示以及测地函数完全一致。我们通过特殊箭图中的簇突变序列实现了辫子群变换。我们证明了归一化量子传输矩阵的群组关系，并且作为副产品，得到了半经典极限下的Goldman括号。证明了任意(循环或非循环)有向平面网络传输矩阵的量子代数关系，以纪念伟大的数学家和人物Boris Dubrovin。
Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of thegroupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables forandin terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations forvia sequences of cluster mutations in the special-quiver. We prove the groupoid relations for normalized quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network.Dedicated to the memory of a great mathematician and person, Boris Dubrovin.