论文标题
$ \ mathscr {a} = \ Mathscr {u} $用于群集代数的$ g $ -local Systems的群集代数
$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems
论文作者
论文摘要
对于有限维的简单谎言代数$ \ mathfrak {g} $承认非平凡的微度表示和一个连接的标记的表面$σ$，至少有两个标记点和没有刺穿，我们证明了群集代数$ \ nathscr {a} _} _ { $（\ Mathfrak {g}，σ）$与上层群集代数$ \ Mathscr {u} _ {\ Mathfrak {g}，σ} $重合。证明是基于以下事实：函数$ \ MATHCAL {o}（\ Mathcal {a}^\ times_ {g，σ}）$ a Moduli twisted twisted $ g $ local Systems的$σ$上的模量空间是由[IO20]中引入的Wilson Lines of Matrix系列生成的$σ$。作为一个应用程序，我们证明了muller-type skein代理$ \ mathscr {s} _ {\ mathfrak {g}}，σ}，σ} [\ partial^{ - 1} $ [muller，iy23，iy23，iy22] \ Mathfrak {sl} _3，$或$ \ Mathfrak {sp} _4 $与群集代数$ \ Mathscr {a} _ {\ Mathfrak {g}，σ} $是同构。
For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $Σ$ with at least two marked points and no punctures, we prove that the cluster algebra $\mathscr{A}_{\mathfrak{g},Σ}$ associated with the pair $(\mathfrak{g},Σ)$ coincides with the upper cluster algebra $\mathscr{U}_{\mathfrak{g},Σ}$. The proof is based on the fact that the function ring $\mathcal{O}(\mathcal{A}^\times_{G,Σ})$ of the moduli space of decorated twisted $G$-local systems on $Σ$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras $\mathscr{S}_{\mathfrak{g}, Σ}[\partial^{-1}]$ [Muller,IY23,IY22] for $\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3,$ or $\mathfrak{sp}_4$ are isomorphic to the cluster algebras $\mathscr{A}_{\mathfrak{g}, Σ}$.
