The quantum dilogarithm and representations quantum cluster varieties

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.Comment: Dedicated to David Kazhdan for his 60th birthday. The final version. To appear in Inventiones Math. The last Section of the previous versions was removed, and will become a separate pape