Universal geometric cluster algebras from surfaces
Universal geometric cluster algebras from surfaces
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DOI：
10.1090/s0002-9947-2014-06156-4
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发表时间：
2012-09
期刊：
影响因子：
--
通讯作者：
Nathan Reading
中科院分区：
文献类型：
--
作者：
Nathan Reading
交换矩阵B上的普适几何聚类代数是B上由系数专门化相关的几何聚类代数范畴中的普适对象。（继先前关于普遍几何簇代数的论文之后，我们扩展了几何簇代数的定义，相对于最初给出的定义Fomin和Zelevinsky。）通称对象与通称b的突变扇F_B密切相关。本文考虑由标记曲面产生的通称几何簇代数和簇代数的突变扇。我们确定了标记表面的两个关键特性：曲线分离特性和零缠结特性。后者的性质意味着前者。证明了除一次穿孔曲面外的所有标记曲面的曲线分离性，得到了这些曲面的有理部分F_B的构造。我们证明了一个较小的曲面族的零缠结性质，并用它来构造这些曲面的通用几何系数。
A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given Fomin and Zelevinsky.) The universal objects are closely related to a fan F_B called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of F_B for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces, and use it to construct universal geometric coefficients for these surfaces.