On certain realizations of the q-deformed exterior differential calculus

We investigate two particular realizations of a q-deformed differential cal culus at q being a primitive root of unity, q(N) = 1. Particular attention is paid to the Z(3)-graded case N = 3. First we construct an analogue of th e exterior differential calculus on a manifold, then we introduce a discret e realization of such a calculus on generalized Clifford algebras. Finally, combining both constructions, we discuss a Z(N)-graded generalization of g auge theory.