Note on Character Varieties and Cluster Algebras

Symmetry Integrability and Geometry-Methods and Applications

article

Kazuhiro Hikami¹ 

[1] Faculty of Mathematics, Kyushu University

关键词: Lie algebras;    coadjoint representation;    contact forms;    Frobenius Lie algebras;    Cartan class;   

DOI  :  10.3842/SIGMA.2019.003

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown