Quantum Teichmüller space from quantum plane

Abstract

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 the 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.

41 pages, 9 figures

References in corpus (1)

• The quantum dilogarithm and representations quantum cluster varieties

Cited by in corpus (7)

• The pentagon relation and incidence geometry
• The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization
• Ratio coordinates for higher Teichmüller spaces
• Phase constants in the Fock-Goncharov quantum cluster varieties
• Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
• A quantization of moduli spaces of 3-dimensional gravity
• Finite dimensional quantum Teichmüller space from the quantum torus at root of unity