The pentagon relation for the quantum dilogarithm and quantized M_{0,5}

Abstract

We introduce and study a Schwarz space S in the space of functions on the real line. It is a module over the algebra L of regular functions on the (modular double of the) non-commutative q-deformation of the moduli space of configurations of 5 cyclically ordered points on the projective line. The algebra L has an order five automorphism corresponding to the cyclic shift of the points. The quantum dilogarithm gives rise to an automorphism of the space Schwarz S intertwining the automorphism of L. This easily implies the pentagon relation for the quantum dilogarithm function. The triple (L, S, the automorphism) is the quantized moduli space of configurations of 5 points on the projective line. It is the simplest example of a quantized cluster X-variety.

12 pages. To appear in the Progress in Mathematics volume (Birkhauser) dedicated to the memory of Alexander Reznikov. The second version is hopefully the final one. It clarifies, corrects and expands the first one

References in corpus (5)

• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• The quantum dilogarithm and representations quantum cluster varieties
• Cluster ensembles, quantization and the dilogarithm
• Quantization of Teichmüller spaces and the quantum dilogarithm

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