Representations of D_q(k [x])

The algebra of quantum differential operators on graded algebras was introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and the second author have identified this algebra of quantum differential operators on the polynomial algebra with coefficients in an algebraically closed field of characteristic zero. It contains the first Weyl algebra and the quantum Weyl algebra as its subalgebras. In this paper we classify irreducible weight modules over the algebra of quantum differential operators on the polynomial algebra. Some classes of indecomposable modules are constructed in the case of positive characteristic and q root of unity.