Unistructurality of cluster algebras from surfaces without punctures
Unistructurality of cluster algebras from surfaces without punctures
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发表时间：
2018-09
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通讯作者：
V'eronique Bazier-Matte;Pierre-Guy Plamondon
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作者：
V'eronique Bazier-Matte;Pierre-Guy Plamondon
一个簇代数是单结构的，如果它的簇变量的集合决定了它的簇和种子。证明了所有的簇代数都是单结构的。在本文中，我们表明，任何集群代数所产生的三角形的一个标记的表面上没有穿孔是单结构的。我们的证明依赖于一个被称为手镯基的正基的存在，以及绞链关系。我们还证明了，从一个不相交的联盟的箭图定义的集群代数是单结构的当且仅当集群代数定义的连通分支的箭图是单结构的。
A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis, and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural.