Quantum Teichmüller space from the quantum plane

• Published in: Duke mathematical journal 2012-02, Vol.161 (2), p.305-366
• Main Authors: Frenkel, Igor B., Kim, Hyun Kyu
• Format: Article
• Language: English
• Subjects: 16T05; 17B37; 20G42; 30F60; 30Fxx; 32G15; 57T05; 81R50; 82B23; Hopf algebras and their applications [See also 16S40; Moduli of Riemann surfaces; Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20; Quantum groups (quantized function algebras) and their representations [See also 16T20; Quantum groups and related algebraic methods [See also 16T20; Teichmüller theory [See also 14H15; Teichmüller theory [See also 32G15]
• ISSN: 0012-7094; 1547-7398
• DOI: 10.1215/00127094-1507390

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of 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 arises 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 show that the quantum universal Teichmüller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichmüller space.