DRINFELD-SOKOLOV REDUCTION FOR DIFFERENCE-OPERATORS AND DEFORMATIONS OF W-ALGEBRAS - I - THE CASE OF VIRASORO ALGEBRA

We propose a q-difference version of the Drinfeld-Sokolov reduction sc heme, which gives us q-deformations of the classical W-algebras by red uction from Poisson-Lie loop groups. We consider in detail the case of SL2. The nontrivial consistency conditions fix the choice of the clas sical r-matrix defining the Poisson-Lie structure on the loop group LS L2, and this leads to a new elliptic classical r-matrix. The reduced P oisson algebra coincides with the deformation of the classical Virasor o algebra previously defined in [19]. We also consider a discrete anal ogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra.