TY - JOUR
T1 - Note on character varieties and cluster algebras
AU - Hikami, Kazuhiro
N1 - Funding Information:
The author would like to thank Thang Le for communications during Workshop “Low-Dimensional Topology and Number Theory” at Mathematisches Forschungsinstitut Oberwolfach in August 2017. He thanks the organizers for invitation. Thanks are also to the speakers of “Geometry of Moduli Spaces and Integrable Systems” at Gakushuin University, Tokyo, in September 2017. This work is supported in part by JSPS KAKENHI Grant Number JP16H03927, JP17K05239, JP17K18781, JP16H02143.
Publisher Copyright:
© 2019, Institute of Mathematics. All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
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U2 - 10.3842/SIGMA.2019.003
DO - 10.3842/SIGMA.2019.003
M3 - Article
AN - SCOPUS:85068124552
SN - 1815-0659
VL - 15
JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
M1 - 003
ER -