Exterior calculus with d3 = 0 on two-dimensional quantum plane

In this paper, we construct the algebra of differential forms with exterior differential satisfying d3 = 0 on the two-dimensional quantum plane assuming that the homomorphism defining first-order differential calculus is linear in variables. Assuming d2 ≠ 0, we introduce the second-order differentials d2xi. The commutation relations between the generators xi, dxi, and d2xi of the algebra of differential forms, among dxi, and among d2xi, as well as between noncommutative derivatives with generators, are found. The consistency conditions are described