Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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