Extensions and contractions of the Lie algebra of q-pseudodifferential symbols on the circle

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. A nontrivial embedding of the Virasoro algebra into (a completion of) q-pseudodifferential symbols is proposed. We describe q-analogs of the KP and KdV-hierarchies admitting an infinite number of conserved charges as well as q-deformed Gelfand-Dickey structures. (C) 1997 Academic Press