Title: Note on character varieties and cluster algebras,
Abstract: 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.