Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths
Cluster Algebras and Triangulated Surfaces
Part II: Lambda Lengths
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簇代数和三角曲面第二部分：Lambda 长度
DOI：
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发表时间：
2012
影响因子：
1.9
通讯作者：
D. Thurston
中科院分区：
文献类型：
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作者：
S. Fomin;D. Thurston
对于任何其底层组合数据可以由带有标记点的有界表面编码的簇代数，我们根据表面的合适的装饰 Teichmueller 空间构造一个几何实现。在几何方面，这需要将每个内部标记点处的表面打开为额外的测地线边界分量。在代数方面，它依赖于非标准化簇代数的概念和热带 lambda 长度的机制。
我们的模型允许任意选择系数，这转化为对表面上的一系列整体叠片的选择。它提供了簇变量的内在解释，即表面上弧的重整化 lambda 长度。交换关系以叠片的剪切坐标表示，并被解释为 lambda 长度的广义托勒密关系。
这种方法为我们之前论文的主要结构结果提供了替代证明，消除了表面上不必要的假设。
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths.
Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths.
This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.