Matrix formulae and skein relations for cluster algebras from surfaces
Matrix formulae and skein relations for cluster algebras from surfaces
复制标题
曲面簇代数的矩阵公式和绞线关系
DOI：
10.1093/imrn/rns118
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
Gregg Musiker;L. Williams
本文研究了与有边曲面（S，M）相关的主系数为A（S，M）的聚类代数，是作者与Schiffler [MSW2]合著的一篇论文的结语。给定曲面上任意（广义）弧或环——有或没有自交——我们用PSL_2(R)的元素积来关联A（S，M）的（分数场）的一个元素。我们直接证明了我们的弧和环的矩阵公式与[MSW， MSW2]中给出的弧和环的组合公式在匹配方面是一致的。最后，我们用矩阵公式证明了与弧和环相关的聚类代数元素的串结关系。我们的矩阵公式和绞结关系推广了Fock和Goncharov [FG1， FG2， FG3]在无系数情况下的先前工作。本文的结果将在[MSW2]中使用，以表明弧和环的某些集合包含a （S，M）的向量空间基。
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fraction field of) A(S,M), using products of elements of PSL_2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW, MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [FG1, FG2, FG3], who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to show that certain collections of arcs and loops comprise a vector-space basis for A(S,M).