On a graded q-differential algebra

Given an associative unital ℤN-graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying the graded q-Leibniz rule and dN = 0. We apply this construction to a reduced quantum plane and study the exterior calculus on a reduced quantum plane induced by the N -differential of the graded q-differential algebra. Making use of the higher order differentials dkx induced by the N -differential d we construct an analogue of an algebra of differential forms with exterior differential satisfying dN = 0. © 2006 Taylor & Francis Group, LLC.