Ratio coordinates for higher Teichmuller spaces
- 주제(기타) Mathematics
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 SPRINGER HEIDELBERG
- 발행년도 2016
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000162028
- 본문언어 영어
- Published As http://dx.doi.org/10.1007/s00209-015-1607-4
초록/요약
We define new coordinates for Fock-Goncharov's higher Teichmuller 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 -space and the -space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of and , together with Poisson structures. We consider new coordinates for higher Teichmuller spaces given as ratios of the coordinates of the -space for , which are generalizations of Kashaev's ratio coordinates in the case . Using Kashaev's quantization for , 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 , and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.
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