QUANTUM GROUP-STRUCTURE AND LOCAL-FIELDS IN THE ALGEBRAIC APPROACH TO2D GRAVITY

This review contains a summary of the work by J.-L. Gervais and the au thor on the operator approach to 2d gravity. Special emphasis is place d on the construction of focal observables - the Liouville exponential s and the Liouville field itself - and the underlying algebra of chira l vertex operators. The double quantum group structure arising from th e presence of two screening charges is discussed and the generalized a lgebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version, of the classical group-th eoretic representation of the Liouville fields by Leznov and Saveliev.