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 d² = 0 as in the usual case. The commutation relations for the generalized differentials ensuring the desired property can be put into the cyclic form a₁a₂a₃...a_N = q a_Na₁a₂...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 × N matrix algebras, or as generalized differential calculus on manifolds. A generalization of gauge theories based on such differential calculus is briefly discussed.