论文标题
链接补充，八面体分解和量子$ \ mathfrak {sl} _2 $上的双曲线结构
Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$
论文作者
论文摘要
双曲线结构（等效地，主$ \ operatorAtorname {psl} _2（\ mathbb c）$ - 与链路补充的$捆绑包）可以通过使用八面体分解来描述代数，该分解将理想的三角剖分分配给链接的任何图。分解（与任何理想的三角剖分一样）给出了一组形状参数的粘合方程，其溶液是双曲线结构。我们表明，这些方程与kac-de concini量子组的一定呈现密切相关，$ \ mathcal {u} _q（\ mathfrak {sl} _2）$在$ q =ξ$的群集代数方面均以$ q =ξ$的统一根。具体而言，我们确定了八面体分解的形状参数的比率，其中心字符$ \ mathcal {u}_ξ（\ Mathfrak {sl} _2）$。已知这些字符上的量子编织与$ \ operatotorname {sl} _2（\ Mathbb c）$ - 链接补充上的捆绑包密切相关，我们的工作为此结构提供了几何视角。
Hyperbolic structures (equivalently, principal $\operatorname{PSL}_2(\mathbb C)$-bundles with connection) on link complements can be described algebraically by using the octahedral decomposition, which assigns an ideal triangulation to any diagram of the link. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations are closely related to a certain presentation of the Kac-de Concini quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ in terms of cluster algebras at $q = ξ$ a root of unity. Specifically, we identify ratios of the shape parameters of the octahedral decomposition with central characters of $\mathcal{U}_ξ(\mathfrak{sl}_2)$. The quantum braiding on these characters is known to be closely related to $\operatorname{SL}_2(\mathbb C)$-bundles on link complements, and our work provides a geometric perspective on this construction.
