Integrable representations of toroidal Lie algebras co-ordinatized by rational quantum tori

For any positive integers d, n with d>1 and n>1, we fix an n by n complex matrix q=(qij) satisfying qii=1 and with all qij roots of unity. Let be the universal central extension of the Lie subalgebra sld(Cq) of gld(Cq) with trace in [Cq,Cq], where Cq is the rational quantum torus associated to q, and let be the Lie algebra by adding the n degree derivations to with respect to the n non-commuting variables in Cq. The Lie algebra , called the toroidal Lie algebra co-ordinatized by the rational quantum torus Cq, has an n-dimensional center C. In this paper, we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non-zero center action over the toroidal Lie algebra .