论文标题
darboux坐标用于符号群和集群代数
Darboux coordinates for symplectic groupoid and cluster algebras
论文作者
论文摘要
使用fock - goncharov较高的teichmüller空间变量，我们得出darboux坐标表示代表$ \ Mathcal a_n $ gropsoid的一般符号叶子的条目，并在更一般的较高符号叶片中，通过补充$ $ rixrix $ rixrix $ rixrix $ rigix $ rigix $ rix complyement opplyemectic sequient。获得的结果与先前获得的泊松和$ \ MATHCAL A_3 $和$ \ MATHCAL A_4 $的poisson和量子表示的量子表示非常完美。我们通过特殊$ \ Mathbb a_n $ -quiver中的群集突变序列代表$ \ Mathcal A_n $的Braid-Group转换。我们证明了量子运输矩阵的群体固定关系，作为副产品，在半经典限制下获得了高盛支架。
Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid 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 $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.
