Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy namely the q-KdV and q-Boussinesq integrable systems. We present also the q-generalization of the conformal transformations of the currents and discuss the primarity condition of the fields by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.