OPERATOR COPRODUCT-REALIZATION OF QUANTUM GROUP TRANSFORMATIONS IN 2-DIMENSIONAL GRAVITY .1.

A simple connection between the universal R matrix of U-q(sl(2)) (for spins 1/2 and J) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct ac tion on the covariant operators. It allows us to derive the expected q uantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of the U-q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained b y showing on a general ground that the link between the algebra of fie ld transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extended U-q(sl(2 )) algebra, which includes the Liouville zero-mode momentum in a non-t rivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry U-q(sl(2)) circle dot U-(q over ca p)(sl(2)) related to the presence of both of the screening charges of 2D gravity.