Z(n)-graded differential calculus

We investigate the properties of differential algebras generated by an operator d satisfying the property d(N) = 0 instead of d(2) = 0 as in the usual case. The commutation relations for the generalized differentials ensuring the desired property can be put into the cyclic form a(1) a(2) a(3) ... a(N) = q a(N) a(1) a(2) ... a(N-1), where q is a primitive N-th root of unity. Examples of realizations of such differential algebras are given, either in the space of Z(N)-graded N x N matrix algebras, or as generalized differential calculus on manifolds. A generalization of gauge theories based on such differential calculus is briefly discussed.