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

TL;DR

This paper explores the pentagon relation for the quantum dilogarithm within the framework of a quantized moduli space of five points, revealing new automorphisms and structure in quantum cluster varieties.

Contribution

It introduces a Schwarz space module over a non-commutative algebra related to the moduli space of five points, establishing the pentagon relation for the quantum dilogarithm.

Findings

Automorphism of the Schwarz space intertwines with the algebra automorphism.

Derivation of the pentagon relation for the quantum dilogarithm.

Identification of the structure as a quantized cluster X-variety.

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.