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Ratio coordinates for higher Teichmuller spaces

Title
Ratio coordinates for higher Teichmuller spaces
Authors
Kim, Hyun Kyu
Ewha Authors
김현규
Issue Date
2016
Journal Title
MATHEMATISCHE ZEITSCHRIFT
ISSN
0025-5874JCR Link

1432-1823JCR Link
Citation
MATHEMATISCHE ZEITSCHRIFT vol. 283, no. 1-2, pp. 469 - 513
Publisher
SPRINGER HEIDELBERG
Indexed
SCI; SCIE; SCOPUS WOS
Document Type
Article
Abstract
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.
DOI
10.1007/s00209-015-1607-4
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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