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.

32 pages, 4 figures

References in corpus (7)

• The quantum dilogarithm and representations quantum cluster varieties
• Cluster integrable systems, q-Painleve equations and their quantization
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• A duality map for quantum cluster varieties from surfaces
• Quantum Holonomies from Spectral Networks and Framed BPS States
• DAHA and skein algebra on surface: double-torus knots
• Algebras of quantum monodromy data and decorated character varieties

Cited by in corpus (2)

• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Okamoto's symmetry on the representation space of the sixth Painlevé equation