Differential operators on monomial curves

Let k be an algebraically closed field of characteristic 0, let Gamma subset of or equal to N-0 be a numerical semigroup, and let A = k[Gamma] be the corresponding semigroup algebra. We give an explicit description of the ring D = D(A) of k-linear differential operators on A, the associated graded ring Gr D(A) and the module of derivations Der(k) (A). We also classify all graded D-modules which are finitely generated torsion-free A-modules of rank 1, considered as modules over the sub-ring A subset of or equal to D(A). (C) 2003 Elsevier Science (USA). All rights reserved.