Note on Character Varieties and Cluster Algebras

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.