| Quantization of the Teichmüller space of a surface, achieved by Kashaev and by Chekhov-Fock(-Goncharov) in somewhat different ways, yields a family of projective representations of the mapping class group of the surface. We derive these projective representations of Kashaev from tensor products of a single canonical representation of a rather basic Hopf algebra Bqq˜, the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also identify the quantum Teichmüller space with a space of intertwiners of Bqq˜, which in particular provides the realization of the quantum universal Teichmüller space as an infinite tensor product of the canonical representation naturally indexed by rational numbers including infinity. Meanwhile, it is known that the quantum universal Teichmüller space yields projective representations of the Prolemy-Thompson group T, which is a universal version of the mapping class groups. The extension class of the central extension T&d14; of T induced by these projective representations coming from the Chekhov-Fock(-Goncharov) quantization has been computed by Funar and Sergiescu to be 12 times the Euler class χ. We compute the explicit presentation of the central extension of T induced by Kashaev quantization instead, and show that its extension class is 6χ. We also give a direct graphical proof of the isomorphism between this new central extension and the group T♯ab , the relative abelianization of the braided Ptolemy-Thompon group T♯ of Funar and Kapoudjian. |
