  We construct cocycles on the Lie algebra of pseudo- and q-pseudodifferential
symbols of one variable and on their close relatives: the sine-algebra and the
Poisson algebra on two-torus. A ``quantum'' Godbillon-Vey cocycle on
(pseudo)-differential operators appears in this construction as a natural
generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We
describe a nontrivial embedding of the Virasoro algebra into (a completion of)
q-pseudodifferential symbols, and propose q-analogs of the KP and
KdV-hierarchies admitting an infinite number of conserved charges.
