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.
Keywords
Cite
@article{arxiv.0706.4054,
title = {The pentagon relation for the quantum dilogarithm and quantized M_{0,5}},
author = {A. B. Goncharov},
journal= {arXiv preprint arXiv:0706.4054},
year = {2007}
}