论文标题
量子痕迹和所述绞线代数的嵌入到量子托里
Quantum traces and embeddings of stated skein algebras into quantum tori
论文作者
论文摘要
刺穿的边界表面（或等效地表面）的陈述的绞线代数是对未标记表面的众所周知的Kauffman支架绞线代数的概括，可以被视为量子范围的特殊线性$ \ narccal $ \ nathcal $ \ nathcal {o} _} _ {q^2} $ nof af af a n big a big，我们表明，用两种不同的方式将带有非空边界界面边界边界边界边界边界边界边界的锯齿代数嵌入量子托里。可以将第一个嵌入方式视为根据增强的Teichmüller空间的剪切坐标表达封闭曲线轨迹的地图的量化，并且是Bonahon-Wong的量子痕迹图的升降。可以将第二个嵌入方式视为地图的量化，以装饰的Teichmüller空间的lambda长度坐标表示了封闭曲线的痕迹，并且是Muller量子痕迹图的扩展。我们解释了两个量子跟踪图之间的关系。我们还表明，Muller的量子群集代数等于所述的绞线代数的简化版本。作为应用程序，我们表明，所述的Skein代数是有序有限生成的Noetherian域，并计算其Gelfand-Kirillov尺寸。
The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group $\mathcal{O}_{q^2}(SL_2)$ from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong's quantum trace map. The second embedding can be considered as a quantization of the map expresses the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller's quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension.
