Note on character varieties and cluster algebras

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Abstract

We use Bonahon-Wong’s trace map to study character varieties of the oncepunctured 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 con uences 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.

Original languageEnglish
Article number3
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume15
DOIs
Publication statusPublished - Jan 1 2019

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Character Variety
Cluster Algebra
Cubic Surface
Poisson Algebra
Quantum Algebra
Poisson Structure
Automorphism
Triangulation
Hilbert
Quantization
Torus
Mutation
Correspondence
Trace

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

Cite this

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