Skein algebras and cluster algebras of marked surfaces
Skein algebras and cluster algebras of marked surfaces
复制标题
标记曲面的绞丝代数和簇代数
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
G. Muller
中科院分区：
文献类型：
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作者：
G. Muller
本文定义了几个代数相关联的一个有向曲面$S$的边界上的一个有限集的标志点。第一个是绞代数$Sk_q（S）$，它由曲面中的链环所张成，这些链环允许端点在标记点上，模若干个局部定义的关系。乘积由链接的叠加给出。给出了该代数的一个基，以及几个代数结果.
当S可三角化时，可以定义量子团簇代数A_q（S）$和量子上团簇代数U_q（S）$。这些代数来自S的三角剖分以及它们之间的基本移动。
本文给出了天然包裹体$A_q（S）$到$Sk_q^o（S）$到$U_q（S）$的关系，其中$Sk_q^o（S）$是$Sk_q（S）$的一个确定的Ore局部化。当$S$的每个分量中至少有两个标记点时，这些夹杂物被加强到相等，在$Sk_q^o（S）$上呈现量子簇结构。
证明这些等式的方法对于证明其它类的簇代数的A_q=U_q具有潜在的意义。作为这一事实的证明，本文给出了无圈簇代数的Aq = Uq的一个新证明
This paper defines several algebras associated to an oriented surface $S$ with a finite set of marked points on the boundary. The first is the skein algebra $Sk_q(S)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results.
When $S$ is triangulable, the quantum cluster algebra $A_q(S)$ and quantum upper cluster algebra U_q(S) can be defined. These are algebras coming from the triangulations of S and the elementary moves between them.
Natural inclusions $A_q(S)$ into $Sk_q^o(S)$ into $U_q(S)$ are shown, where $Sk_q^o(S)$ is a certain Ore localization of $Sk_q(S)$. When $S$ has at least two marked points in each component, these inclusions are strengthened to equality, exhibiting a quantum cluster structure on $Sk_q^o(S)$.
The method for proving these equalities has potential to show $A_q=U_q$ for other classes of cluster algebras. As a demonstration of this fact, a new proof is given that $A_q=U_q$ for acyclic cluster algebras