On a graded q-differential algebra

We construct the graded q-differential algebra on a ZN -graded algebra by means of a graded q-commutator. We apply this construction to a reduced quantum plane and study the first order differential calculus on a reduced quantum plane induced by the N -differential of the graded q-differential algebra. 1 Graded q-differential algebra In this section given a ZN -graded algebra we construct the graded q-differential algebra. Let us remind the definition of a graded q-differential algebra ([1]). A unital associative algebra is said to be a graded q-differential algebra (q ∈ C, q 6= 1) if it is a ZN -graded (or Z-graded) algebra endowed with the linear mapping d of degree +1 satisfying the graded q-Leibniz rule and d = 0 in the case when q is a primitive N -th root of unity. The linear mapping d is called an N -differential of a graded q-differential algebra. Let A be an associative unital Z (or ZN)-graded algebra over the complex numbers C and A ⊂ A be the subspace of homogeneous elements of a grading k. The grading of a homogeneous element w will be denoted by |w|, which means that if w ∈ A then |w| = k. Let q be a complex number such that q 6= 1. The q-commutator of two homogeneous elements w,w ∈ A is defined by the formula [w,w]q = ww ′ − q ′ww. (1) Using the associativity of an algebra A and the property |ww| = |w| + |w| of its graded structure it is easy to show that for any homogeneous elements w,w, w ∈ A the q-commutator has the property [w,ww]q = [w,w ]qw ′′ + q ′w[w,w]q. (2)