Classification of irreducible bounded weight modules over the derivation Lie algebras of quantum tori

Let d>1d>1 be an integer, q=(qij)d×dq=(qij)d×d a d×dd×d complex matrix satisfying qii=1qii=1, qij=qji−1 with all qijqij being roots of unity. Let CqCq be the rational quantum torus algebra associated with q  , and Der(Cq)Der(Cq) its derivation Lie algebra. In this paper, we give a complete classification of irreducible bounded weight modules over Der(Cq)Der(Cq). They turn out to be irreducible sub-quotients of Der(Cq)Der(Cq)-module Vα(V,W)Vα(V,W) for a finite dimensional irreducible gldgld-module V  , a finite dimensional Γ-graded-irreducible glNglN-module W  , and α∈Cdα∈Cd where the integer N is uniquely determined by q.