Generalized q-deformed Gelfand-Dickey structures on the group of q-pseudodifference operators

We define the q-deformed Gelfand-Dickey bracket on the space of q-pseudodifference symbols which agrees with the Poisson Virasoro algebra of E.Frenkel and N.Reshetikhin and its generalizations and prove its uniqueness (in a natural class of quadratic Poisson structures). The associated hierarchies of nonlinear q-difference equations are also constructed.