Extensions and Contractions of the Lie Algebra ofq-Pseudodifferential Symbols on the Circle

We construct cocycles on the Lie algebra of pseudo- andq-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 describeq-analogs of the KP and KdV-hierarchies admitting an infinite number of conserved charges as well asq-deformed Gelfand–Dickey structures.