THE QUANTUM GROUP-STRUCTURE OF 2D GRAVITY AND MINIMAL MODELS .2. THE GENUS-ZERO CHIRAL BOOTSTRAP

The chiral operator-algebra of the quantum-group-covariant operators (of vertex type) is completely worked out by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. This completes earlier articles along the same line. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the authors (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to infinity in a suitable way. ne introduction of quantum-group-covariant operators in the three dimensional picture gives a generalization of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work and a previous parallel article gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.