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

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

2007-06-27

arxiv.org/abs/0706.4054v2

Mathematics

Quantum Algebra

12 pages. To appear in the Progress in Mathematics volume (Birkhauser) dedicated to the memory of Alexander Reznikov. The seco

Scientific paper

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.

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