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We propose a q-difference version of the Drinfeld-Sokolov reduction scheme,which gives us q-deformations of the classical W-algebras by reduction fromPoisson-Lie loop groups. We consider in detail the case of SL(2). Thenontrivial consistency conditions fix the choice of the classical r-matrixdefining the Poisson-Lie structure on the loop group LSL(2), and this leads toa new elliptic classical r-matrix. The reduced Poisson algebra coincides withthe deformation of the classical Virasoro algebra previously defined inq-alg/9505025. We also consider a discrete analogue of this Poisson algebra. Inthe second part (q-alg/9702016) the construction is generalized to the case ofan arbitrary semisimple Lie algebra.Comment: 29 pages, AMSLATE

Drinfeld-Sokolov Reduction for Difference Operators and Deformations of ${\cal W}$ -Algebras¶I. The Case of Virasoro Algebra