The Quantum Teichmüller Space as a Noncommutative Algebraic Object

We consider the quantum Teichmüller space of the punctured surface introduced by ChekhovFock-Kashaev, and formalize it as a noncommutative deformation of the space of algebraic functions on the Teichmüller space of the surface. In order to apply it in 3-dimensional topology, we put more attention to the details involving small surfaces. Let S be an oriented surface of finite topological type, with at least one puncture. A quantization of the Teichmüller space T(S) of S was developed by L. Chekhov and V. Fock [7, 8, 5] and, independently, by R. Kashaev [11] (see also [14]) as an approach to quantum gravity in 2+1 dimensions. This is a deformation of the C–algebra of functions on the usual Teichmüller space T(S) of S, depending on a parameter ~, in such a way that the linearization of this deformation at ~ = 0 corresponds to the Weil-Petersson Poisson structure on T(S). In this paper, we develop a slightly different version of this quantization, which has a more algebraic flavor. It essentially is the image under the exponential map of the quantization of Chekhov-Fock-Kashaev. The original quantization was expressed in terms of self-adjoint operators on Hilbert spaces and made strong use of the holomorphic function