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SIGMA 5 (2009), 015, 20 pages arXiv:0902.1302
https://doi.org/10.3842/SIGMA.2009.015
Contribution to the Special Issue on Kac-Moody Algebras and Applications
The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichmüller Space
Armen G. Sergeev
Steklov Mathematical Institute, 8 Gubkina Str., 119991 Moscow, Russia
Received July 29, 2008, in final form February 05, 2009; Published online February 08, 2009
Abstract
In the first part of the paper we describe the complex
geometry of the universal Teichmüller space T, which
may be realized as an open subset in the complex Banach space of
holomorphic quadratic differentials in the unit disc. The quotient
S of the diffeomorphism group of the circle modulo
Möbius transformations may be treated as a smooth part of T. In the second part we consider the quantization of
universal Teichmüller space T. We explain first how to
quantize the smooth part S by embedding it into a
Hilbert-Schmidt Siegel disc. This quantization method, however,
does not apply to the whole universal Teichmüller space T, for its quantization we use an approach, due to Connes.
Key words:
universal Teichmüller space; quasisymmetric homeomorphisms; Connes quantization.
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