  We exhibit direct relations between the modular double of U_q(sl(2,R)) and
the quantum Teichmueller theory. Explicit representations for the fusion- and
braiding operations of the quantum Teichmueller theory are immediate
consequences. Our results include a simplified derivation of the Clebsch-Gordan
decomposition for the principal series of representation of the modular double
of U_q(sl(2,R)).
