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Quantum Teichmüller space from the quantum plane. (English) Zbl 1271.30020
Summary: We derive the quantum Teichmüller space, previously constructed by R. M. Kashaev [Lett. Math. Phys. 43, No. 2, 105–115 (1998; Zbl 0897.57014)] and by L. O. Chekhov and V. V. Fock [Theor. Math. Phys. 120, No. 3, 1245–1259 (1999); translation from Teor. Mat. Fiz. 120, No. 3, 511–528 (1999; Zbl 0986.32007)], 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.

MSC:
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations
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