Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2206.15226 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917867679645696 |
|---|---|
| author | Asaka, Takeru Ishibashi, Tsukasa Kano, Shunsuke |
| author_facet | Asaka, Takeru Ishibashi, Tsukasa Kano, Shunsuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_15226 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Earthquake theorem for cluster algebras of finite type Asaka, Takeru Ishibashi, Tsukasa Kano, Shunsuke Geometric Topology Combinatorics 51H25 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. |
| title | Earthquake theorem for cluster algebras of finite type |
| topic | Geometric Topology Combinatorics 51H25 |
| url | https://arxiv.org/abs/2206.15226 |