  We construct a discrete quantum version of the Drinfeld-Sokolov
correspondence for the sine-Gordon system. The classical version of this
correspondence is a birational Poisson morphism between the phase space of the
discrete sine-Gordon system and a Poisson homogeneous space. Under this
correspondence, the commuting higher mKdV vector fields correspond to the
action of an Abelian Lie algebra. We quantize this picture (1) by quantizing
this Poisson homogeneous space, together with the action of the Abelian Lie
algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the
quantum analogues of the integrals of motion generating the mKdV vector fields,
and (4) by constructing an algebra morphism taking one commuting family of
derivations to the other one.
