论文标题
$ \ mathfrak {Sp} _4 $
Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$
论文作者
论文摘要
继续进行我们以前的工作[IY21]（Arxiv：2101.00643），$ \ \ \ Mathfrak {sl} _3 $ - case，我们介绍了一个skein algebra $ \ mathscr {s} s} _ {\ m mathfrak {\ mathfrak {Sp}在特殊点的特定“锁定”绞线的标志表面$σ$上，并研究了其群集性质。我们还介绍了一个自然的$ \ Mathbb {z} _q $ - form $ \ mathscr {s} _ {\ mathfrak {sp} _4，σ}^{\ MathBB {z} _q} _Q} _Q} _q} \ subset \ subset \ subset \ mathscr {s}}系数环$ \ Mathcal {r} $的$ \ Mathscr {s} _ {\ Mathfrak {sp} _4，σ}^q $包括量子整数$ [2] _Q $的倒数。我们证明其边界 - 元素化$ \ mathscr {s} _ {\ mathfrak {sp} _4，σ}^{\ mathbb {z} _q} _q} [\ partial^{ - 1}] $包含$ \ mathscr {a}^q _ {\ mathfrak {sp} _4，σ} $量化Moduli Space $ \ Mathcal {A} _ {sp_4，σ}^\ times $的函数环。此外，我们以与[IY21]相似的方式获得了隆布人表达式的阳性表达式（ARXIV：2101.00643）。我们还提出了fomin-pylyavksyy [fp16]（arxiv：1210.1888）的精神的集群变量的特征，就$ \ mathfrak {sp} _4 $ -webs而言，并给出了无限的许多支撑示例。
Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $Σ$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,Σ}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,Σ}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,Σ}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,Σ}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral.
