Representations of the Lie algebra of derivations for quantum torus

Let Der(Cq) be the derivation Lie algebra of the quantum torus Cq. We construct a functor from gld-modules to Der(Cq)-modules, which generalizes the constructions obtained by Shen [Scientia Sinica A 29 (1986) 570], Larsson [Internat. J. Modern Phys. A 7 (1992) 6493], and Rao [J. Algebra 182 (1996) 401]. We also give a complete description of the structure of the Der(Cq)-modules when all the entries of the quantum torus matrix q=(qij)d×d are roots of unity.