Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra

First we briefly describe the link between the Virasoro algebra and the free scalar field on a two-dimensional space-time given as a standard commutative cylinder, and in the Euclidean version on a complex plane. The field-theoretical model generalized then to the noncommutative cylinder leads to discrete time-evolution. Its Euclidean version is shown to be equivalent to a model on a complex q-plane. There is a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra suggested earlier, which describes the symmetry of the theory. The problems with the supersymmetric extension of the model on a noncommutative super-space are briefly discussed. PACS: 03.70