关于特征标簇与簇代数的注记
Note on character varieties and cluster algebras
摘要：
我们使用Bonahon-Wong的迹图来研究一次穿孔环面和四次穿孔球面的特征变量。我们阐明了与曲面理想三角剖分相关的簇代数的关系，并证明了曲面上环的Goldman Poisson代数可以从簇代数的Poisson结构中恢复出来。它还表明，簇突变给出的字符品种的自同构。受Chekhov-Mazzocco-Rubtsov工作的启发，我们从簇代数的角度重新研究了球面上的穿刺汇流，并得到了van der Put-Saito基于Riemann-Hilbert对应构造的相关仿射三次曲面.进一步研究了特征标簇的量子化问题。
英文摘要：
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on sphere from cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studied are quantizations of character varieties by use of quantum cluster algebra.
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