Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{sl}}_3$
Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{sl}}_3$
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${mathfrak{sl}}_3$ 的未穿孔表面的绞纱和簇代数
DOI：
10.1007/s00209-023-03208-7
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发表时间：
2022
影响因子：
0.8
通讯作者：
Yuasa Wataru
中科院分区：
文献类型：
--
作者：
Ishibashi Tsukasa;Yuasa Wataru
对于未穿孔的标记曲面，我们考虑一个包含-网的skein代数，在标记点处具有边界skein关系。本文在skew-分数域中构造了一个量子簇代数，它将上的装饰局部系统模空间上的簇结构量子化，证明了该簇代数包含边界局部skein代数作为子代数，并且它们的自然结构（如分次和某些群作用）是一致的.我们还给出了一个算法来计算一个给定的网络在某些集群的洛朗表达式和讨论的积极性系数。特别地，我们证明了环中的手镯和沿有向简单环沿着的手镯具有正系数的Laurent表达式，从而得到量子GS-泛正Laurent多项式.
For an unpunctured marked surface, we consider a skein algebraconsisting of-webs onwith the boundary skein relations at marked points. We construct a quantum cluster algebrainside the skew-fieldof fractions, which quantizes the cluster-structure on the moduli spaceof decorated-local systems on. We show that the cluster algebracontains the boundary-localized skein algebraas a subalgebra, and their natural structures, such as gradings and certain group actions, agree with each other. We also give an algorithm to compute the Laurent expressions of a given-web in certain clusters and discuss the positivity of coefficients. In particular, we show that the bracelets and the bangles along an oriented simple loop inhave Laurent expressions with positive coefficients, hence give rise to quantum GS-universally positive Laurent polynomials.