Irreducible modules over the derivation algebras of rational quantum tori

Let g be the derivation Lie algebra of a rational quantum torus Cq where q=(qij)d×d and all qij are roots of unity. In Lin and Tan (2004) [LT], , a class of uniformly bounded irreducible weight modules over g were constructed, which generalized the construction given by Shen (1986) [Sh], , Larsson (1992) [L], , and Eswara Rao (1996) [E], . These modules look mysterious because of their very unclear weight sets. In this note, we use very simple methods with a very short proof to re-establish all (even stronger) results in Lin and Tan (2004) [LT] and make these mysterious modules crystal clear. A necessary and sufficient condition for two such modules to be isomorphic is also given.