  We consider a field theoretical model on the noncommutative cylinder which
leads to a discrete-time evolution. Its Euclidean version is shown to be
equivalent to a model on the complex $q$-plane. We reveal a direct link between
the model on a noncommutative cylinder and the deformed Virasoro algebra
constructed earlier on an abstract mathematical background. As it was shown,
the deformed Virasoro generators necessarily carry a second index (in addition
to the usual one), whose meaning, however, remained unknown. The present field
theoretical approach allows one to ascribe a clear meaning to this second
index: its origin is related to the noncommutativity of the underlying
space-time. The problems with the supersymmetric extension of the model on a
noncommutative super-space are briefly discussed.
