QUANTUM GROUP GENERATORS IN CONFORMAL FIELD-THEORY

Two approaches to the construction of symmetry generators for the quan tum group U-q(l(2))in conformal field theory are presented, in the con crete context of 2d gravity. The first works with an extension of the physical phase space and has been successfully applied already to WZW theory. We show that the result can be used also for Liouville theory and related models by employing Hamiltonian reduction. The second is b ased on a completely new idea and realizes the quantum group symmetry intrinsically, on the physical phase space alone.