Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 003, 32 pages      arXiv:1711.03379      https://doi.org/10.3842/SIGMA.2019.003

Note on Character Varieties and Cluster Algebras

Kazuhiro Hikami
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received July 25, 2018, in final form January 10, 2019; Published online January 20, 2019

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.

Key words: cluster algebra; character variety; Painlevé equations; Goldman Poisson algebra.

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