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

Understand

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.