The irreducible modules for the derivations of the rational quantum torus

Let CqCq be the quantum torus associated with the d×dd×d matrix q=(qij)q=(qij), where qijqij are roots of unity with qii=1qii=1 and qij−1=qji for all 1≤i,j≤d1≤i,j≤d. Let Der(Cq)Der(Cq) be the Lie algebra of all the derivations of CqCq. In this paper we define the Lie algebra Der(Cq)⋉CqDer(Cq)⋉Cq and classify its irreducible modules with finite dimensional weight spaces. These modules under certain conditions turn out to be of the form V⊗CqV⊗Cq, where V   is a finite dimensional irreducible gldgld-module.