TY - JOUR

T1 - Ratio coordinates for higher Teichmüller spaces

AU - Kim, Hyun Kyu

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We define new coordinates for Fock–Goncharov’s higher Teichmüller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group G. Some additional data on the boundary leads to two closely related moduli spaces, the (Formula presented.) -space and the (Formula presented.) -space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of (Formula presented.) and (Formula presented.) , together with Poisson structures. We consider new coordinates for higher Teichmüller spaces given as ratios of the coordinates of the (Formula presented.) -space for (Formula presented.) , which are generalizations of Kashaev’s ratio coordinates in the case (Formula presented.). Using Kashaev’s quantization for (Formula presented.) , we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for (Formula presented.) , and for completeness we also give a full proof of the presentation of Kashaev’s groupoid of decorated ideal triangulations.

AB - We define new coordinates for Fock–Goncharov’s higher Teichmüller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group G. Some additional data on the boundary leads to two closely related moduli spaces, the (Formula presented.) -space and the (Formula presented.) -space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of (Formula presented.) and (Formula presented.) , together with Poisson structures. We consider new coordinates for higher Teichmüller spaces given as ratios of the coordinates of the (Formula presented.) -space for (Formula presented.) , which are generalizations of Kashaev’s ratio coordinates in the case (Formula presented.). Using Kashaev’s quantization for (Formula presented.) , we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for (Formula presented.) , and for completeness we also give a full proof of the presentation of Kashaev’s groupoid of decorated ideal triangulations.

UR - http://www.scopus.com/inward/record.url?scp=84951791300&partnerID=8YFLogxK

U2 - 10.1007/s00209-015-1607-4

DO - 10.1007/s00209-015-1607-4

M3 - Article

AN - SCOPUS:84951791300

VL - 283

SP - 469

EP - 513

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -