论文标题
地震定理，用于有限类型的集群代数
Earthquake theorem for cluster algebras of finite type
论文作者
论文摘要
我们为有限类型的群集代数引入了Thurston地震图的集群代数概括，我们称之为\ emph {cluster {集群地震映射}。它是由粘合指数图来定义的，该图指数图是按照理想弧的地震建模的。我们证明了地震定理的类似物，该类似物指出，群集地震图在$ \ Mathbb {r}^\ mathrm {trop} $ - 和$ \ m athbb {r} _ {> 0} _ {> 0} $ - cluse $ mathcal $ \ mathcal covle vartial的空间之间给出同构的同态。对于$ a_n $和$ d_n $的类型的人，群集地震地图确实分别恢复了标记磁盘和曾经是刻有标记磁盘的地震图。此外，我们研究了集群地震图的某些渐近行为，这些行为引起了Fock-Goncharov风扇的“连续变形”。
We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb{R}^\mathrm{trop}$- and $\mathbb{R}_{>0}$-valued points of the cluster $\mathcal{X}$-variety. For those of type $A_n$ and $D_n$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock--Goncharov fan.
